From a0e31ce884eb5e7cc7f2708fa158be6ea61612bc Mon Sep 17 00:00:00 2001 From: unalmis Date: Tue, 27 Aug 2024 20:48:28 -0400 Subject: [PATCH 01/44] Fix missed things with manual merge in commit 8d8aa78 --- desc/integrals/bounce_integral.py | 1 - desc/integrals/bounce_utils.py | 1 - tests/test_integrals.py | 5 ++--- 3 files changed, 2 insertions(+), 5 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 7429a9bcb..bb2240a97 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -596,7 +596,6 @@ class Bounce1D: coordinate systems may be used as this operation reduces to a surface integral, which is invariant to the order of summation). - The DESC coordinate system is related to field-line-following coordinate systems by a relation whose solution is best found with Newton iteration. There is a unique real solution to this equation, so Newton iteration is a diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 3cd7905e2..34dab1b54 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -654,7 +654,6 @@ def interp_to_argmin( exp(max(g)) << exp(``upper_sentinel``). Don't make too large or numerical resolution is lost. - Warnings -------- Recall that if g is small then the effect of β is reduced. diff --git a/tests/test_integrals.py b/tests/test_integrals.py index e3d328acf..d2f595080 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1150,11 +1150,10 @@ def drift_analytic(data): Numerically computed ``data["cvdrift"]` and ``data["gbdrift"]`` normalized by some scale factors for this unit test. These should be fed to the bounce integration as input. - pitch : jnp.ndarray + pitch : jnp.ndarray Shape (P, ). Pitch values used. - """ B = data["|B|"] / data["Bref"] B0 = np.mean(B) @@ -1336,7 +1335,7 @@ def test_binormal_drift_bounce1d(self): self._test_bounce_autodiff( bounce, - TestBounce1DPoints.drift_num_integrand, + TestBounce1D.drift_num_integrand, f=f, weight=np.ones(zeta.size), ) From dbd80a55baf9be973f10f6e44e26072bf944f773 Mon Sep 17 00:00:00 2001 From: unalmis Date: Tue, 17 Sep 2024 16:01:37 -0400 Subject: [PATCH 02/44] Reviving fourier bounce pull request --- desc/integrals/basis.py | 55 ++++++++++------------- desc/integrals/bounce_integral.py | 75 ++++++++++++++++--------------- desc/integrals/interp_utils.py | 27 +++++++++-- desc/utils.py | 12 +++++ tests/test_integrals.py | 4 +- 5 files changed, 99 insertions(+), 74 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index b9ec3c78c..0ccad22c5 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -8,6 +8,7 @@ from desc.backend import dct, flatnonzero, idct, irfft, jnp, put, rfft from desc.integrals.interp_utils import ( _filter_distinct, + _subtract_first, cheb_from_dct, cheb_pts, chebroots_vec, @@ -29,23 +30,6 @@ ) -def _subtract(c, k): - """Subtract ``k`` from first index of last axis of ``c``. - - Semantically same as ``return c.copy().at[...,0].add(-k)``, - but allows dimension to increase. - """ - c_0 = c[..., 0] - k - c = jnp.concatenate( - [ - c_0[..., jnp.newaxis], - jnp.broadcast_to(c[..., 1:], (*c_0.shape, c.shape[-1] - 1)), - ], - axis=-1, - ) - return c - - @partial(jnp.vectorize, signature="(m),(m)->(m)") def _in_epigraph_and(is_intersect, df_dy_sign, /): """Set and epigraph of function f with the given set of points. @@ -162,12 +146,11 @@ def __init__(self, f, domain=(-1, 1), lobatto=False): @staticmethod def _fast_transform(f, lobatto): N = f.shape[-1] - c = rfft( + return rfft( dct(f, type=2 - lobatto, axis=-1) / (N - lobatto), axis=-2, norm="forward", ) - return c @staticmethod def nodes(M, N, L=None, domain=(-1, 1), lobatto=False): @@ -204,8 +187,9 @@ def nodes(M, N, L=None, domain=(-1, 1), lobatto=False): coords = (jnp.atleast_1d(L), x, y) else: coords = (x, y) - coords = list(map(jnp.ravel, jnp.meshgrid(*coords, indexing="ij"))) - coords = jnp.column_stack(coords) + coords = jnp.column_stack( + list(map(jnp.ravel, jnp.meshgrid(*coords, indexing="ij"))) + ) return coords def evaluate(self, M, N): @@ -424,7 +408,7 @@ def intersect2d(self, k=0.0, eps=_eps): Sign of ∂f/∂y (x, yᵢ). """ - c = _subtract(_chebcast(self.cheb, k), k) + c = _subtract_first(_chebcast(self.cheb, k), k) # roots yᵢ of f(x, y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y) - k(x) y = chebroots_vec(c) assert y.shape == (*c.shape[:-1], self.N - 1) @@ -432,9 +416,9 @@ def intersect2d(self, k=0.0, eps=_eps): # Intersects must satisfy y ∈ [-1, 1]. # Pick sentinel such that only distinct roots are considered intersects. y = _filter_distinct(y, sentinel=-2.0, eps=eps) - is_intersect = (jnp.abs(y.imag) <= eps) & (jnp.abs(y.real) <= 1.0) - # Ensure y is in domain of arcos; choose 1 because kernel probably cheaper. - y = jnp.where(is_intersect, y.real, 1.0) + is_intersect = (jnp.abs(y.imag) <= eps) & (jnp.abs(y.real) < 1.0) + # Ensure y is in differentiable domain of arcos: (-1, 1). + y = jnp.where(is_intersect, y.real, 0) # TODO: Multipoint evaluation with FFT. # Chapter 10, https://doi.org/10.1017/CBO9781139856065. @@ -473,7 +457,7 @@ def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): z1, z2 : (jnp.ndarray, jnp.ndarray) Shape broadcasts with (..., *self.cheb.shape[:-2], num_intersect). ``z1`` and ``z2`` are intersects satisfying ∂f/∂y <= 0 and ∂f/∂y >= 0, - respectively. The points are grouped and ordered such that the straight + respectively. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of f. """ @@ -500,7 +484,9 @@ def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): # this, for those subset of pitch values the integrations will be done in # the hypograph of |B|, which will yield zero. If in far future decide to # not ignore this, note the solution is to disqualify intersects within - # ``_eps`` from ``domain[-1]``. + # ``_eps`` from ``domain[-1]``. Edit: For differentiability, we cannot + # consider intersects at boundary of Chebyshev polynomial. Again, cases + # where this would be incorrect have measure zero. is_z1 = (df_dy_sign <= 0) & is_intersect is_z2 = (df_dy_sign >= 0) & _in_epigraph_and(is_intersect, df_dy_sign) @@ -519,7 +505,8 @@ def _check_shape(self, z1, z2, k): # Ensure pitch batch dim exists and add back dim to broadcast with wells. k = atleast_nd(self.cheb.ndim - 1, k)[..., jnp.newaxis] # Same but back dim already exists. - z1, z2 = atleast_nd(self.cheb.ndim, z1, z2) + z1 = atleast_nd(self.cheb.ndim, z1) + z2 = atleast_nd(self.cheb.ndim, z2) # Cheb has shape (..., M, N) and others # have shape (K, ..., W) errorif(not (z1.ndim == z2.ndim == k.ndim == self.cheb.ndim)) @@ -533,7 +520,7 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): z1, z2 : jnp.ndarray Shape must broadcast with (*self.cheb.shape[:-2], W). ``z1`` and ``z2`` are intersects satisfying ∂f/∂y <= 0 and ∂f/∂y >= 0, - respectively. The points are grouped and ordered such that the straight + respectively. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of f. k : jnp.ndarray Shape must broadcast with *self.cheb.shape[:-2]. @@ -560,8 +547,8 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): # Ensure l axis exists for iteration in below loop. cheb = atleast_nd(3, self.cheb) - mask, z1, z2, f_midpoint = atleast_3d_mid(mask, z1, z2, f_midpoint) - err_1, err_2, err_3 = atleast_2d_end(err_1, err_2, err_3) + mask, z1, z2, f_midpoint = map(atleast_3d_mid, (mask, z1, z2, f_midpoint)) + err_1, err_2, err_3 = map(atleast_2d_end, (err_1, err_2, err_3)) for l in np.ndindex(cheb.shape[:-2]): for p in range(k.shape[0]): @@ -610,6 +597,7 @@ def plot1d( hlabel=r"$z$", vlabel=r"$f$", show=True, + include_legend=True, ): """Plot the piecewise Chebyshev series. @@ -641,6 +629,8 @@ def plot1d( Vertical axis label. show : bool Whether to show the plot. Default is true. + include_legend : bool + Whether to include the legend in the plot. Default is true. Returns ------- @@ -666,7 +656,8 @@ def plot1d( ) ax.set_xlabel(hlabel) ax.set_ylabel(vlabel) - ax.legend(legend.values(), legend.keys(), loc="lower right") + if include_legend: + ax.legend(legend.values(), legend.keys(), loc="lower right") ax.set_title(title) plt.tight_layout() if show: diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 1826791a9..b65d2a7eb 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -104,8 +104,8 @@ def _transform_to_desc(grid, f): # After GitHub issue #1034 is resolved, we should pass in the previous # θ(α) coordinates as an initial guess for the next coordinate mapping. # Perhaps tell the optimizer to perturb the coefficients of the -# |B|(α, ζ) directly? Maybe auto diff to see change on |B|(θ, ζ) -# and hence stream functions. Not sure how feasible... +# |B|(α, ζ) directly? think perturbing alpha is equivalent to perturbing +# lambda. Not sure if possible.. # TODO: Allow multiple starting labels for near-rational surfaces. # can just concatenate along second to last axis of cheb, but will @@ -115,12 +115,12 @@ def _transform_to_desc(grid, f): class Bounce2D: """Computes bounce integrals using two-dimensional pseudo-spectral methods. - The bounce integral is defined as ∫ f(ℓ) dℓ, where + The bounce integral is defined as ∫ f(λ, ℓ) dℓ, where dℓ parameterizes the distance along the field line in meters, - f(ℓ) is the quantity to integrate along the field line, - and the boundaries of the integral are bounce points ζ₁, ζ₂ s.t. λ|B|(ζᵢ) = 1, - where λ is a constant proportional to the magnetic moment over energy - and |B| is the norm of the magnetic field. + f(λ, ℓ) is the quantity to integrate along the field line, + and the boundaries of the integral are bounce points ℓ₁, ℓ₂ s.t. λ|B|(ℓᵢ) = 1, + where λ is a constant defining the integral proportional to the magnetic moment + over energy and |B| is the norm of the magnetic field. For a particle with fixed λ, bounce points are defined to be the location on the field line such that the particle's velocity parallel to the magnetic field is zero. @@ -290,11 +290,17 @@ def __init__( quad : (jnp.ndarray, jnp.ndarray) Quadrature points xₖ and weights wₖ for the approximate evaluation of an integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). Default is 32 points. + For weak singular integrals, use ``chebgauss2`` from + ``desc.integrals.quad_utils``. + For strong singular integrals, use ``leggauss``. automorphism : (Callable, Callable) or None The first callable should be an automorphism of the real interval [-1, 1]. The second callable should be the derivative of the first. This map defines a change of variable for the bounce integral. The choice made for the automorphism will affect the performance of the quadrature method. + For weak singular integrals, use ``None``. + For strong singular integrals, use ``automorphism_sin`` from + ``desc.integrals.quad_utils``. Bref : float Optional. Reference magnetic field strength for normalization. Lref : float @@ -421,17 +427,16 @@ def _L(self): """int: Number of flux surfaces to compute on.""" return self._B.cheb.shape[0] - def bounce_points(self, pitch, num_well=None): + def bounce_points(self, pitch_inv, num_well=None): """Compute bounce points. Parameters ---------- - pitch : jnp.ndarray - Shape (P, L). - λ values to evaluate the bounce integral at each field line. λ(ρ) is - specified by ``pitch[...,ρ]`` where in the latter the labels ρ are - interpreted as the index into the last axis that corresponds to that field - line. If two-dimensional, the first axis is the batch axis. + pitch_inv : jnp.ndarray + Shape (M, L, P). # TODO: right now set up is (P, L). + 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by + ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted + as the indices that correspond to that field line. num_well : int or None Specify to return the first ``num_well`` pairs of bounce points for each pitch along each field line. This is useful if ``num_well`` tightly @@ -451,9 +456,9 @@ def bounce_points(self, pitch, num_well=None): epigraph of |B|. """ - return self._B.intersect1d(1 / jnp.atleast_2d(pitch), num_well) + return self._B.intersect1d(jnp.atleast_2d(pitch_inv), num_well) - def check_bounce_points(self, z1, z2, pitch, plot=True, **kwargs): + def check_bounce_points(self, z1, z2, pitch_inv, plot=True, **kwargs): """Check that bounce points are computed correctly. Parameters @@ -463,12 +468,11 @@ def check_bounce_points(self, z1, z2, pitch, plot=True, **kwargs): ζ coordinates of bounce points. The points are grouped and ordered such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. - pitch : jnp.ndarray - Shape (P, L). - λ values to evaluate the bounce integral at each field line. λ(ρ) is - specified by ``pitch[...,ρ]`` where in the latter the labels ρ are - interpreted as the index into the last axis that corresponds to that field - line. If two-dimensional, the first axis is the batch axis. + pitch_inv : jnp.ndarray + Shape (M, L, P). # TODO: right now set up is (P, L). + 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by + ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted + as the indices that correspond to that field line. plot : bool Whether to plot stuff. kwargs : dict @@ -483,22 +487,21 @@ def check_bounce_points(self, z1, z2, pitch, plot=True, **kwargs): kwargs.setdefault("klabel", r"$1/\lambda$") kwargs.setdefault("hlabel", r"$\zeta$") kwargs.setdefault("vlabel", r"$\vert B \vert$") - self._B.check_intersect1d(z1, z2, 1 / pitch, plot, **kwargs) + self._B.check_intersect1d(z1, z2, pitch_inv, plot, **kwargs) - def integrate(self, pitch, integrand, f, weight=None, num_well=None): - """Bounce integrate ∫ f(ℓ) dℓ. + def integrate(self, pitch_inv, integrand, f, weight=None, num_well=None): + """Bounce integrate ∫ f(λ, ℓ) dℓ. - Computes the bounce integral ∫ f(ℓ) dℓ for every specified field line + Computes the bounce integral ∫ f(λ, ℓ) dℓ for every specified field line for every λ value in ``pitch``. Parameters ---------- - pitch : jnp.ndarray - Shape (P, L). - λ values to evaluate the bounce integral at each field line. λ(ρ) is - specified by ``pitch[...,ρ]`` where in the latter the labels ρ are - interpreted as the index into the last axis that corresponds to that field - line. If two-dimensional, the first axis is the batch axis. + pitch_inv : jnp.ndarray + Shape (M, L, P). # TODO: right now set up is (P, L). + 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by + ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted + as the indices that correspond to that field line. integrand : callable The composition operator on the set of functions in ``f`` that maps the functions in ``f`` to the integrand f(ℓ) in ∫ f(ℓ) dℓ. It should accept the @@ -514,7 +517,7 @@ def integrate(self, pitch, integrand, f, weight=None, num_well=None): weight : jnp.ndarray Shape (L, 1, m, n). If supplied, the bounce integral labeled by well j is weighted such that - the returned value is w(j) ∫ f(ℓ) dℓ, where w(j) is ``weight`` + the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` interpolated to the deepest point in the magnetic well. Use the method ``self.reshape_data`` to reshape the data into the expected shape. num_well : int or None @@ -535,9 +538,9 @@ def integrate(self, pitch, integrand, f, weight=None, num_well=None): Last axis enumerates the bounce integrals. """ - pitch = jnp.atleast_2d(pitch) - z1, z2 = self.bounce_points(pitch, num_well) - result = self._integrate(z1, z2, pitch, integrand, f) + pitch_inv = jnp.atleast_2d(pitch_inv) + z1, z2 = self.bounce_points(pitch_inv, num_well) + result = self._integrate(z1, z2, pitch_inv, integrand, f) errorif(weight is not None, NotImplementedError) return result diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 3dbc5b14a..efb3b916b 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -20,6 +20,7 @@ # TODO: Boyd's method 𝒪(N²) instead of Chebyshev companion matrix 𝒪(N³). # John P. Boyd, Computing real roots of a polynomial in Chebyshev series # form through subdivision. https://doi.org/10.1016/j.apnum.2005.09.007. +# This is likely the bottleneck. chebroots_vec = jnp.vectorize(chebroots, signature="(m)->(n)") @@ -143,10 +144,11 @@ def harmonic_vander(x, M): # TODO: For inverse transforms, do multipoint evaluation with FFT. # FFT cost is 𝒪(M N log[M N]) while direct evaluation is 𝒪(M² N²). # Chapter 10, https://doi.org/10.1017/CBO9781139856065. -# Right now we just do an MMT with the Vandermode matrix. -# Multipoint is likely better than using NFFT to evaluate f(xq) given fourier -# coefficients because evaluation points are quadratically packed near edges as -# required by quadrature to avoid runge. NFFT is only approximation anyway. +# Right now we do an MMT with the Vandermode matrix. +# Multipoint is likely better than using NFFT (for strong singular bounce +# integrals) to evaluate f(xq) given fourier coefficients because evaluation +# points are quadratically packed near edges for efficient quadrature. For +# weak singularities (e.g. effective ripple) NFFT should work well. # https://github.com/flatironinstitute/jax-finufft. @@ -451,6 +453,23 @@ def polyval_vec(*, x, c): # TODO: Eventually do a PR to move this stuff into interpax. +def _subtract_first(c, k): + """Subtract ``k`` from first index of last axis of ``c``. + + Semantically same as ``return c.copy().at[...,0].add(-k)``, + but allows dimension to increase. + """ + c_0 = c[..., 0] - k + c = jnp.concatenate( + [ + c_0[..., jnp.newaxis], + jnp.broadcast_to(c[..., 1:], (*c_0.shape, c.shape[-1] - 1)), + ], + axis=-1, + ) + return c + + def _subtract_last(c, k): """Subtract ``k`` from last index of last axis of ``c``. diff --git a/desc/utils.py b/desc/utils.py index 72dd10f97..63375075b 100644 --- a/desc/utils.py +++ b/desc/utils.py @@ -742,6 +742,18 @@ def atleast_nd(ndmin, ary): return jnp.array(ary, ndmin=ndmin) if jnp.ndim(ary) < ndmin else ary +def atleast_3d_mid(ary): + """Like np.atleast_3d but if adds dim at axis 1 for 2d arrays.""" + ary = jnp.atleast_2d(ary) + return ary[:, jnp.newaxis] if ary.ndim == 2 else ary + + +def atleast_2d_end(ary): + """Like np.atleast_2d but if adds dim at axis 1 for 1d arrays.""" + ary = jnp.atleast_1d(ary) + return ary[:, jnp.newaxis] if ary.ndim == 1 else ary + + PRINT_WIDTH = 60 # current longest name is BootstrapRedlConsistency with pre-text diff --git a/tests/test_integrals.py b/tests/test_integrals.py index ca4e7e6d9..f08b7e215 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1635,7 +1635,7 @@ def integrand_den(B, pitch): normalization = -np.sign(data["psi"]) * data["Bref"] * data["a"] ** 2 drift_numerical_num = bounce.integrate( - pitch=pitch[:, np.newaxis], + pitch_inv=pitch[:, np.newaxis], integrand=integrand_num, f=Bounce2D.reshape_data( grid, @@ -1645,7 +1645,7 @@ def integrand_den(B, pitch): num_well=1, ) drift_numerical_den = bounce.integrate( - pitch=pitch[:, np.newaxis], + pitch_inv=pitch[:, np.newaxis], integrand=integrand_den, f=[], num_well=1, From 29fa91398ceecd591a83a11f773c546b572032aa Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 18 Sep 2024 05:37:31 -0400 Subject: [PATCH 03/44] Progressing --- desc/integrals/basis.py | 95 +++--- desc/integrals/bounce_integral.py | 509 +++++++++++++++++------------- desc/integrals/bounce_utils.py | 67 ++-- desc/integrals/interp_utils.py | 12 +- tests/test_integrals.py | 1 + 5 files changed, 392 insertions(+), 292 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 0ccad22c5..82db1983b 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -7,6 +7,7 @@ from desc.backend import dct, flatnonzero, idct, irfft, jnp, put, rfft from desc.integrals.interp_utils import ( + _eps, _filter_distinct, _subtract_first, cheb_from_dct, @@ -18,6 +19,7 @@ irfft_non_uniform, ) from desc.integrals.quad_utils import bijection_from_disc, bijection_to_disc +from desc.io import IOAble from desc.utils import ( atleast_2d_end, atleast_3d_mid, @@ -83,8 +85,11 @@ def _in_epigraph_and(is_intersect, df_dy_sign, /): def _chebcast(cheb, arr): - # Input should not have rightmost dimension of cheb that iterates coefficients, - # but may have additional leftmost dimension for batch operation. + """Add leftmost axis to ``cheb`` depending on ``arr.ndim``. + + Input ``arr`` should not have rightmost dimension of cheb that iterates + coefficients, but may have additional leftmost dimension for batch operation. + """ errorif( jnp.ndim(arr) > cheb.ndim, NotImplementedError, @@ -94,7 +99,7 @@ def _chebcast(cheb, arr): return cheb if jnp.ndim(arr) < cheb.ndim else cheb[jnp.newaxis] -class FourierChebyshevBasis: +class FourierChebyshevBasis(IOAble): """Fourier-Chebyshev series. f(x, y) = ∑ₘₙ aₘₙ ψₘ(x) Tₙ(y) @@ -112,11 +117,11 @@ class FourierChebyshevBasis: Fourier spectral resolution. N : int Chebyshev spectral resolution. + domain : (float, float) + Domain for y coordinates. lobatto : bool Whether ``f`` was sampled on the Gauss-Lobatto (extrema-plus-endpoint) instead of the interior roots grid for Chebyshev points. - domain : (float, float) - Domain for y coordinates. """ @@ -249,14 +254,14 @@ def compute_cheb(self, x): Chebyshev coefficients αₙ(x=``x``) for f(x, y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). """ - # Always add new axis to broadcast against Chebyshev coefficients. + # Add axis to broadcast against Chebyshev coefficients. x = jnp.atleast_1d(x)[..., jnp.newaxis] cheb = cheb_from_dct(irfft_non_uniform(x, self._c, self.M, axis=-2), axis=-1) assert cheb.shape[-2:] == (x.shape[-2], self.N) return ChebyshevBasisSet(cheb, self.domain) -class ChebyshevBasisSet: +class ChebyshevBasisSet(IOAble): """Chebyshev series. { fₓ | fₓ : y ↦ ∑ₙ₌₀ᴺ⁻¹ aₙ(x) Tₙ(y) } @@ -276,8 +281,6 @@ class ChebyshevBasisSet: """ - _eps = min(jnp.finfo(jnp.array(1.0).dtype).eps * 1e2, 1e-10) - def __init__(self, cheb, domain=(-1, 1)): """Make Chebyshev series basis from given coefficients. @@ -384,7 +387,7 @@ def eval1d(self, z, cheb=None): assert f.shape == z.shape return f - def intersect2d(self, k=0.0, eps=_eps): + def intersect2d(self, k=0.0, *, eps=_eps): """Coordinates yᵢ such that f(x, yᵢ) = k(x). Parameters @@ -417,8 +420,8 @@ def intersect2d(self, k=0.0, eps=_eps): # Pick sentinel such that only distinct roots are considered intersects. y = _filter_distinct(y, sentinel=-2.0, eps=eps) is_intersect = (jnp.abs(y.imag) <= eps) & (jnp.abs(y.real) < 1.0) - # Ensure y is in differentiable domain of arcos: (-1, 1). - y = jnp.where(is_intersect, y.real, 0) + # Ensure y ∈ (-1, 1), i.e. where arccos is differentiable. + y = jnp.where(is_intersect, y.real, 0.0) # TODO: Multipoint evaluation with FFT. # Chapter 10, https://doi.org/10.1017/CBO9781139856065. @@ -434,7 +437,7 @@ def intersect2d(self, k=0.0, eps=_eps): y = bijection_from_disc(y, self.domain[0], self.domain[-1]) return y, is_intersect, df_dy_sign - def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): + def intersect1d(self, k=0.0, *, num_intersect=None, pad_value=0.0): """Coordinates z(x, yᵢ) such that fₓ(yᵢ) = k for every x. Parameters @@ -456,9 +459,9 @@ def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): ------- z1, z2 : (jnp.ndarray, jnp.ndarray) Shape broadcasts with (..., *self.cheb.shape[:-2], num_intersect). - ``z1`` and ``z2`` are intersects satisfying ∂f/∂y <= 0 and ∂f/∂y >= 0, - respectively. The points are ordered and grouped such that the straight - line path between ``z1`` and ``z2`` resides in the epigraph of f. + Coordinates of intersects. The points are ordered and grouped such + that the straight line path between ``z1`` and ``z2`` resides in + the epigraph of f. """ errorif( @@ -468,11 +471,11 @@ def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): f"but got N = {self.N}.", ) - # Add axis to use same k over all Chebyshev series of the piecewise object. + # Add axis to use same k over all Chebyshev series of the piecewise spline. y, is_intersect, df_dy_sign = self.intersect2d( jnp.atleast_1d(k)[..., jnp.newaxis] ) - # Flatten so that last axis enumerates intersects along the piecewise object. + # Flatten so that last axis enumerates intersects along the piecewise spline. y, is_intersect, df_dy_sign = map( flatten_matrix, (self.isomorphism_to_C1(y), is_intersect, df_dy_sign) ) @@ -502,6 +505,7 @@ def intersect1d(self, k=0.0, num_intersect=None, pad_value=0.0): def _check_shape(self, z1, z2, k): """Return shapes that broadcast with (k.shape[0], *self.cheb.shape[:-2], W).""" + assert z1.shape == z2.shape # Ensure pitch batch dim exists and add back dim to broadcast with wells. k = atleast_nd(self.cheb.ndim - 1, k)[..., jnp.newaxis] # Same but back dim already exists. @@ -519,31 +523,36 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): ---------- z1, z2 : jnp.ndarray Shape must broadcast with (*self.cheb.shape[:-2], W). - ``z1`` and ``z2`` are intersects satisfying ∂f/∂y <= 0 and ∂f/∂y >= 0, - respectively. The points are ordered and grouped such that the straight - line path between ``z1`` and ``z2`` resides in the epigraph of f. + Coordinates of intersects. The points are ordered and grouped such + that the straight line path between ``z1`` and ``z2`` resides in + the epigraph of f. k : jnp.ndarray - Shape must broadcast with *self.cheb.shape[:-2]. + Shape must broadcast with self.cheb.shape[:-2]. k such that fₓ(yᵢ) = k. plot : bool - Whether to plot stuff. Default is true. + Whether to plot the piecewise spline and intersects for the given ``k``. kwargs : dict Keyword arguments into ``self.plot``. + Returns + ------- + plots : list + Matplotlib (fig, ax) tuples for the 1D plot of each field line. + """ - assert z1.shape == z2.shape + plots = [] + z1, z2, k = self._check_shape(z1, z2, k) mask = (z1 - z2) != 0.0 z1 = jnp.where(mask, z1, jnp.nan) z2 = jnp.where(mask, z2, jnp.nan) - z1, z2, k = self._check_shape(z1, z2, k) err_1 = jnp.any(z1 > z2, axis=-1) err_2 = jnp.any(z1[..., 1:] < z2[..., :-1], axis=-1) f_midpoint = self.eval1d((z1 + z2) / 2) - assert f_midpoint.shape == z1.shape - err_3 = jnp.any(f_midpoint > k + self._eps, axis=-1) + eps = kwargs.pop("eps", jnp.finfo(jnp.array(1.0).dtype).eps * 10) + err_3 = jnp.any(f_midpoint > k + eps, axis=-1) if not (plot or jnp.any(err_1 | err_2 | err_3)): - return + return plots # Ensure l axis exists for iteration in below loop. cheb = atleast_nd(3, self.cheb) @@ -563,6 +572,10 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): z1=_z1, z2=_z2, k=k[idx], + title=kwargs.pop( + "title", r"Intersects $z$ in epigraph($f$) s.t. $f(z) = k$" + ) + + f", (p,l)={idx}", **kwargs, ) print(" z1 | z2") @@ -570,19 +583,22 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): assert not err_1[idx], "Intersects have an inversion.\n" assert not err_2[idx], "Detected discontinuity.\n" assert not err_3[idx], ( - "Detected f > k in well, implying the straight line path between " - "z1 and z2 is in hypograph(f). Increase spectral resolution.\n" - f"{f_midpoint[idx][mask[idx]]} > {k[idx] + self._eps}" + f"Detected f = {f_midpoint[idx][mask[idx]]} > {k[idx] + _eps} = k" + "in well, implying the straight line path between z1 and z2 is in" + "hypograph(f). Increase spectral resolution.\n" ) idx = (slice(None), *l) if plot: - self.plot1d( - cheb=cheb[l], - z1=z1[idx], - z2=z2[idx], - k=k[idx], - **kwargs, + plots.append( + self.plot1d( + cheb=cheb[l], + z1=z1[idx], + z2=z2[idx], + k=k[idx], + **kwargs, + ) ) + return plots def plot1d( self, @@ -599,7 +615,7 @@ def plot1d( show=True, include_legend=True, ): - """Plot the piecewise Chebyshev series. + """Plot the piecewise Chebyshev series ``cheb``. Parameters ---------- @@ -634,7 +650,8 @@ def plot1d( Returns ------- - fig, ax : matplotlib figure and axes + fig, ax + Matplotlib (fig, ax) tuple. """ fig, ax = plt.subplots() diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index b65d2a7eb..c4a4b0093 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -1,13 +1,16 @@ """Methods for computing bounce integrals (singular or otherwise).""" +import numpy as np from interpax import CubicHermiteSpline, PPoly +from numpy.fft import rfft2 from orthax.legendre import leggauss -from desc.backend import jnp, rfft2 -from desc.integrals.basis import FourierChebyshevBasis +from desc.backend import jnp +from desc.integrals.basis import ChebyshevBasisSet, FourierChebyshevBasis from desc.integrals.bounce_utils import ( _bounce_quadrature, _check_bounce_points, + _check_interp, _set_default_plot_kwargs, bounce_points, get_alpha, @@ -23,96 +26,89 @@ grad_automorphism_sin, ) from desc.io import IOAble -from desc.utils import errorif, flatten_matrix, setdefault, warnif +from desc.utils import atleast_nd, errorif, flatten_matrix, setdefault, warnif -def _transform_to_clebsch(grid, desc_from_clebsch, M, N, B): - """Transform to Clebsch spectral domain. +def _transform_to_desc(grid, f, is_reshaped=False): + """Transform to DESC spectral domain. Parameters ---------- grid : Grid - Tensor-product grid in (ρ, θ, ζ) with uniformly spaced nodes in + Tensor-product grid in (θ, ζ) with uniformly spaced nodes in (2π × 2π) poloidal and toroidal coordinates. - Note that below shape notation defines - L = ``grid.num_rho``, m = ``grid.num_theta``, and n = ``grid.num_zeta``. - desc_from_clebsch : jnp.ndarray - Shape (L * M * N, 3). - DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,domain=FourierBounce.domain)``. - M : int - Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. A good choice is ``m``. If the poloidal stream - function condenses the Fourier spectrum of |B| significantly, then a - larger number may be beneficial. - N : int - Grid resolution in toroidal direction for Clebsch coordinate grid. - Preferably power of 2. - B : jnp.ndarray - |B| evaluated on ``grid``. + f : jnp.ndarray + Function evaluated on ``grid``. Returns ------- - T, B : (FourierChebyshevBasis, FourierChebyshevBasis) + a : jnp.ndarray + Shape (..., grid.num_theta // 2 + 1, grid.num_zeta) + Complex coefficients of 2D real FFT of ``f``. """ - T = FourierChebyshevBasis( - # θ is computed on the optimal nodes in Clebsch space, - # which is a tensor product node set in Clebsch space. - f=desc_from_clebsch[:, 1].reshape(grid.num_rho, M, N), - domain=Bounce2D.domain, - ) - B = FourierChebyshevBasis( - f=interp_rfft2( - # Interpolate to optimal nodes in Clebsch space, - # which is not a tensor product node set in DESC space. - xq=desc_from_clebsch[:, 1:].reshape(grid.num_rho, -1, 2), - f=grid.meshgrid_reshape(B, order="rtz")[:, jnp.newaxis], - # Real fft over poloidal since usually num theta > num zeta. - axes=(-1, -2), - ).reshape(grid.num_rho, M, N), - domain=Bounce2D.domain, - ) - return T, B + if not is_reshaped: + f = grid.meshgrid_reshape(f, "rtz") + # real fft over poloidal since usually m > n + a = rfft2(f, axes=(-1, -2), norm="forward") + assert a.shape[-2:] == (grid.num_theta // 2 + 1, grid.num_zeta) + return a -def _transform_to_desc(grid, f): - """Transform to DESC spectral domain. +def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): + """Transform to Clebsch spectral domain. Parameters ---------- grid : Grid - Tensor-product grid in (θ, ζ) with uniformly spaced nodes in + Tensor-product grid in (ρ, θ, ζ) with uniformly spaced nodes in (2π × 2π) poloidal and toroidal coordinates. - f : jnp.ndarray - Function evaluated on ``grid``. + Note that below shape notation defines + L = ``grid.num_rho``, m = ``grid.num_theta``, and n = ``grid.num_zeta``. + Preferably power of 2 for ``m`` and ``n``. + desc_from_clebsch : jnp.ndarray + Shape (L, M, N, 3). + DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + B : jnp.ndarray + |B| evaluated on ``grid``. Returns ------- - a : jnp.ndarray - Shape (grid.num_rho, grid.num_theta // 2 + 1, grid.num_zeta) - Complex coefficients of 2D real FFT. + T, B : (FourierChebyshevBasis, FourierChebyshevBasis) + Spectral coefficients of θ(α, ζ) and |B|(α, ζ). """ - f = grid.meshgrid_reshape(f, order="rtz") - a = rfft2(f, axes=(-1, -2), norm="forward") - assert a.shape == (grid.num_rho, grid.num_theta // 2 + 1, grid.num_zeta) - return a + domain = (0, 2 * jnp.pi) + M, N = desc_from_clebsch.shape[-3:-1] + # θ is computed on the optimal nodes in Clebsch space, + # which is a tensor product node set in Clebsch space. + T = FourierChebyshevBasis(f=desc_from_clebsch[..., 1], domain=domain) + + if not is_reshaped: + B = grid.meshgrid_reshape(B, "rtz") + # Interpolate to optimal nodes in Clebsch space, + # which is not a tensor product node set in DESC space. + B = interp_rfft2( + xq=desc_from_clebsch[..., 1:].reshape(-1, M * N, 2), + f=B[..., jnp.newaxis, :, :], + axes=(-1, -2), + ).reshape(-1, M, N) + B = FourierChebyshevBasis( + f=B if desc_from_clebsch.ndim == 4 else B.squeeze(axis=0), + domain=domain, + ) + return T, B -# TODO: -# After GitHub issue #1034 is resolved, we should pass in the previous +# TODO: After GitHub issue #1034 is resolved, we should pass in the previous # θ(α) coordinates as an initial guess for the next coordinate mapping. # Perhaps tell the optimizer to perturb the coefficients of the # |B|(α, ζ) directly? think perturbing alpha is equivalent to perturbing # lambda. Not sure if possible.. -# TODO: Allow multiple starting labels for near-rational surfaces. -# can just concatenate along second to last axis of cheb, but will -# do in later pull request since it's not urgent. - -class Bounce2D: +class Bounce2D(IOAble): """Computes bounce integrals using two-dimensional pseudo-spectral methods. The bounce integral is defined as ∫ f(λ, ℓ) dℓ, where @@ -168,13 +164,8 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) fast discrete polynomial transforms (DPT) are implemented via fast transform to Chebyshev then DCT. Although nothing prohibits a direct DPT, we want to rely on existing, optimized libraries. There are other reasons to prefer - Chebyshev series not discussed here. - - Therefore, |B| is interpolated to a Fourier-Chebyshev series in (α, ζ). - The roots of f are computed as the eigenvalues of the Chebyshev companion - matrix. This will later be replaced with Boyd's method: - Computing real roots of a polynomial in Chebyshev series form through - subdivision. https://doi.org/10.1016/j.apnum.2005.09.007. + Chebyshev series not discussed here. Therefore, |B| is interpolated to a + Fourier-Chebyshev series in (α, ζ). Computing accurate series expansions in (α, ζ) coordinates demands particular interpolation points in that coordinate system. Newton iteration @@ -191,12 +182,19 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) Globally convergent root-finding schemes for that basis (at fixed α) are not known. The denominator of a close rational could be absorbed into the coordinate ϕ, but this balloons the frequency, and hence the degree of the - series. Although, because Fourier series may converge faster than Chebyshev, - an alternate strategy that should work is to interpolate |B| to a double - Fourier series in (ϑ, ϕ), then apply bisection methods to find roots of f - with mesh size inversely proportional to the max frequency along the field - line: M ι + N. ``Bounce2D`` does not use that approach because that - root-finding scheme is inferior. + series. + + (Although, because Fourier series may converge faster than Chebyshev, + an alternate strategy is to interpolate |B| to a double Fourier series in + (ϑ, ϕ), then apply bisection methods to find roots of f with mesh size + inversely proportional to the max frequency along the field line: M ι + N. + ``Bounce2D`` does not use that approach because that root-finding scheme is + inferior. Note that if the Chebyshev series convergence is not satisfactory, + an option with faster convergence would be to use a filtered Fourier series + with a non-periodic basis in the Gibbs layer. An example is given at + doi.org/10.1016/j.aml.2006.10.001. Before choosing that option however, + one should test a few change of variables for the Chebyshev interpolation + that may allow earlier truncation of the series without loss of accuracy.) After obtaining the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. @@ -208,50 +206,47 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) the linear algebra libraries of JAX are used. It should be worthwhile to use the inverse non-uniform fast transforms. Fast multipoint methods are preferable because they are exact, but this requires more development work. - Future work may implement these techniques, along with empirical testing of - a few change of variables for the Chebyshev interpolation that may allow - earlier truncation of the series without loss of accuracy. + Future work may implement these techniques. See Also -------- Bounce1D Uses one-dimensional local spline methods for the same task. - An advantage of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on + Advantages of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on which the root-finding must be done to map from DESC to Clebsch coords is - fixed to ``L*M*N``, independent of the number of toroidal transits. + fixed to ``L*M*N``, independent of the number of toroidal transits, and + the function approximation of |B|(ζ) converges faster. Warnings -------- - It is assumed that ζ = ϕ. + It is currently assumed that ζ = ϕ. Attributes ---------- - _B : ChebyshevBasisSet - Set of 1D Chebyshev spectral coefficients of |B| along field line. - {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the - sequence of poloidal coordinates that specify the field line. - _T : ChebyshevBasisSet - Set of 1D Chebyshev spectral coefficients of θ along field line. - {θ_α : ζ ↦ θ(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the - sequence of poloidal coordinates that specify the field line. + required_names : list + Names in ``data_index`` required to compute bounce integrals. """ - domain = (0, 2 * jnp.pi) + required_names = ["B^zeta", "|B|", "iota"] + get_pitch_inv = staticmethod(get_pitch_inv) def __init__( self, grid, data, desc_from_clebsch, - M, - N, - alpha_0=0.0, + # TODO: Allow multiple starting labels for near-rational surfaces. + # think can just concatenate along second to last axis of cheb, but will + # do in later pull request. + alpha=0.0, num_transit=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), Bref=1.0, Lref=1.0, + *, + is_reshaped=False, check=False, **kwargs, ): @@ -259,8 +254,8 @@ def __init__( Notes ----- - Performance may improve significantly - if the spectral resolutions ``M`` and ``N`` are powers of two. + Performance may improve significantly if the spectral + resolutions ``m``, ``n``, ``M``, and ``N`` are powers of two. Parameters ---------- @@ -270,20 +265,13 @@ def __init__( Note that below shape notation defines L = ``grid.num_rho``, m = ``grid.num_theta``, and n = ``grid.num_zeta``. data : dict[str, jnp.ndarray] - Data evaluated on ``grid``. Must include ``FourierBounce.required_names()``. + Data evaluated on ``grid``. + Must include names in ``Bounce2D.required_names``. desc_from_clebsch : jnp.ndarray - Shape (L * M * N, 3). + Shape (L, M, N, 3). DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=FourierBounce.domain)``. - M : int - Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. A good choice is ``m``. If the poloidal stream - function condenses the Fourier spectrum of |B| significantly, then a - larger number may be beneficial. - N : int - Grid resolution in toroidal direction for Clebsch coordinate grid. - Preferably power of 2. - alpha_0 : float + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + alpha : float Starting field line poloidal label. num_transit : int Number of toroidal transits to follow field line. @@ -305,6 +293,12 @@ def __init__( Optional. Reference magnetic field strength for normalization. Lref : float Optional. Reference length scale for normalization. + is_reshaped : bool + Whether the arrays in ``data`` are already reshaped to the expected form of + shape (..., m, n) or (L, m, n). This option can be used to iteratively + compute bounce integrals one flux surface at a time, reducing memory usage + To do so, set to true and provide only those axes of the reshaped data. + Default is false. check : bool Flag for debugging. Must be false for JAX transformations. @@ -323,34 +317,36 @@ def __init__( self._n = grid.num_zeta self._x, self._w = get_quadrature(quad, automorphism) - # Compute global splines. + # Compute spectral coefficients. self._b_sup_z = _transform_to_desc( - grid, - jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, - )[:, jnp.newaxis] + grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped + ) T, B = _transform_to_clebsch( - grid, - desc_from_clebsch, - M, - N, - data["|B|"] / Bref, + grid, desc_from_clebsch, data["|B|"] / Bref, is_reshaped ) # peel off field lines alphas = get_alpha( - alpha_0, - grid.compress(data["iota"]), - num_transit, - period=Bounce2D.domain[-1], + alpha, + iota=( + grid.compress(data["iota"].ravel()) + if data["iota"].size == grid.num_nodes + # assume passed in reshaped data over flux surface + else jnp.atleast_1d(data["iota"][0, 0]) + ), + num_transit=num_transit, + period=2 * jnp.pi, ) + + # Set of 1D Chebyshev spectral coefficients of |B| along field line. + # {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the + # sequence of poloidal coordinates that specify the field line. self._B = B.compute_cheb(alphas) + # Likewise with θ. # Evaluating set of Chebyshev series more efficient than evaluating # Fourier Chebyshev series, so we project θ to Chebyshev series as well. self._T = T.compute_cheb(alphas) assert self._B.M == self._T.M == num_transit - assert self._B.N == self._T.N == N - assert ( - self._B.cheb.shape == self._T.cheb.shape == (grid.num_rho, num_transit, N) - ) + assert self._B.N == self._T.N == desc_from_clebsch.shape[-2] @staticmethod def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): @@ -388,7 +384,7 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): """ if clebsch is None: - clebsch = FourierChebyshevBasis.nodes(M, N, L, Bounce2D.domain) + clebsch = FourierChebyshevBasis.nodes(M, N, L, domain=(0, 2 * jnp.pi)) desc_coords = eq.map_coordinates( coords=clebsch, inbasis=("rho", "alpha", "zeta"), @@ -398,50 +394,48 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): return desc_coords @staticmethod - def required_names(): - """Return names in ``data_index`` required to compute bounce integrals.""" - return ["B^zeta", "|B|", "iota"] - - @staticmethod - def reshape_data(grid, *data): + def reshape_data(grid, *arys): """Reshape ``data`` arrays for acceptable input to ``integrate``. Parameters ---------- grid : Grid Tensor-product grid in (ρ, θ, ζ). - data : jnp.ndarray + arys : jnp.ndarray Data evaluated on grid. Returns ------- - f : list[jnp.ndarray] - List of reshaped arrays which may be given to ``integrate``. + f : jnp.ndarray + Shape (L, M, N). + Reshaped data which may be given to ``integrate``. """ - f = [grid.meshgrid_reshape(d, "rtz")[:, jnp.newaxis] for d in data] - return f + f = [grid.meshgrid_reshape(d, "rtz") for d in arys] + return f if len(f) > 1 else f[0] - @property - def _L(self): - """int: Number of flux surfaces to compute on.""" - return self._B.cheb.shape[0] + @staticmethod + def _swap_pl(f): + """Given shape (L, P, -1) or (P, L, -1) or (P, -1) swap L and P.""" + assert f.ndim <= 3 + return jnp.swapaxes(f, 0, -2) - def bounce_points(self, pitch_inv, num_well=None): + def points(self, pitch_inv, *, num_well=None): """Compute bounce points. Parameters ---------- pitch_inv : jnp.ndarray - Shape (M, L, P). # TODO: right now set up is (P, L). - 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by - ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted + Shape (L, P). + 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by + ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. num_well : int or None Specify to return the first ``num_well`` pairs of bounce points for each pitch along each field line. This is useful if ``num_well`` tightly - bounds the actual number. As a reference, there are typically at most 5 - wells per toroidal transit for a given pitch. + bounds the actual number. As a reference, there are typically 20 wells + per toroidal transit for a given pitch. You can check this by plotting + the field lines with the ``check_points`` method. If not specified, then all bounce points are returned. If there were fewer wells detected along a field line than the size of the last axis of the @@ -450,107 +444,147 @@ def bounce_points(self, pitch_inv, num_well=None): Returns ------- z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (P, L, num_well). - ζ coordinates of bounce points. The points are grouped and ordered such + Shape (L, P, num_well). + ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. + If there were less than ``num_well`` wells detected along a field line, + then the last axis, which enumerates bounce points for a particular field + line and pitch, is padded with zero. + """ - return self._B.intersect1d(jnp.atleast_2d(pitch_inv), num_well) + pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T + # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. + z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) + z1 = self._swap_pl(z1) + z2 = self._swap_pl(z2) + return z1, z2 - def check_bounce_points(self, z1, z2, pitch_inv, plot=True, **kwargs): + def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """Check that bounce points are computed correctly. Parameters ---------- - z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (P, L, num_well). - ζ coordinates of bounce points. The points are grouped and ordered such + z1, z2 : jnp.ndarray + Shape (L, P, num_well). + ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. pitch_inv : jnp.ndarray - Shape (M, L, P). # TODO: right now set up is (P, L). - 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by - ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted + Shape (L, P). + 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by + ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. plot : bool - Whether to plot stuff. + Whether to plot the field lines and bounce points of the given pitch angles. kwargs : dict - Keyword arguments into ``ChebyshevBasisSet.plot1d``. + Keyword arguments into + ``desc/integrals/basis.py::ChebyshevBasisSet.plot1d``. + + Returns + ------- + plots : list + Matplotlib (fig, ax) tuples for the 1D plot of each field line. """ - kwargs.setdefault( - "title", - r"Intersects $\zeta$ in epigraph($\vert B \vert$) s.t. " - r"$\vert B \vert(\zeta) = 1/\lambda$", + return self._B.check_intersect1d( + z1=self._swap_pl(z1), + z2=self._swap_pl(z2), + k=atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T, + plot=plot, + **_set_default_plot_kwargs(kwargs), ) - kwargs.setdefault("klabel", r"$1/\lambda$") - kwargs.setdefault("hlabel", r"$\zeta$") - kwargs.setdefault("vlabel", r"$\vert B \vert$") - self._B.check_intersect1d(z1, z2, pitch_inv, plot, **kwargs) - def integrate(self, pitch_inv, integrand, f, weight=None, num_well=None): + def integrate( + self, + integrand, + pitch_inv, + f=None, + weight=None, + *, + num_well=None, + check=False, + plot=False, + ): """Bounce integrate ∫ f(λ, ℓ) dℓ. - Computes the bounce integral ∫ f(λ, ℓ) dℓ for every specified field line - for every λ value in ``pitch``. + Computes the bounce integral ∫ f(λ, ℓ) dℓ for every field line and pitch. Parameters ---------- - pitch_inv : jnp.ndarray - Shape (M, L, P). # TODO: right now set up is (P, L). - 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by - ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted - as the indices that correspond to that field line. integrand : callable The composition operator on the set of functions in ``f`` that maps the - functions in ``f`` to the integrand f(ℓ) in ∫ f(ℓ) dℓ. It should accept the - arrays in ``f`` as arguments as well as the additional keyword arguments: - ``B`` and ``pitch``. A quadrature will be performed to approximate the - bounce integral of ``integrand(*f,B=B,pitch=pitch)``. - f : list[jnp.ndarray] - Shape (L, 1, m, n). + functions in ``f`` to the integrand f(λ, ℓ) in ∫ f(λ, ℓ) dℓ. It should + accept the arrays in ``f`` as arguments as well as the additional keyword + arguments: ``B`` and ``pitch``. A quadrature will be performed to + approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. + pitch_inv : jnp.ndarray + Shape (L, P). + 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by + ``pitch_inv[ρ]`` where in the latter the labels are interpreted + as the indices that correspond to that field line. + f : list[jnp.ndarray] or jnp.ndarray + Shape (L, m, n). Real scalar-valued (2π × 2π) periodic in (θ, ζ) functions evaluated on the ``grid`` supplied to construct this object. These functions should be arguments to the callable ``integrand``. Use the method ``self.reshape_data`` to reshape the data into the expected shape. weight : jnp.ndarray - Shape (L, 1, m, n). + Shape (L, m, n). If supplied, the bounce integral labeled by well j is weighted such that the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` - interpolated to the deepest point in the magnetic well. Use the method + interpolated to the deepest point in that magnetic well. Use the method ``self.reshape_data`` to reshape the data into the expected shape. num_well : int or None Specify to return the first ``num_well`` pairs of bounce points for each pitch along each field line. This is useful if ``num_well`` tightly - bounds the actual number. As a reference, there are typically at most 5 - wells per toroidal transit for a given pitch. + bounds the actual number. As a reference, there are typically 20 wells + per toroidal transit for a given pitch. You can check this by plotting + the field lines with the ``check_points`` method. If not specified, then all bounce points are returned. If there were fewer wells detected along a field line than the size of the last axis of the returned arrays, then that axis is padded with zero. + check : bool + Flag for debugging. Must be false for JAX transformations. + plot : bool + Whether to plot the quantities in the integrand interpolated to the + quadrature points of each integral. Ignored if ``check`` is false. Returns ------- result : jnp.ndarray - Shape (P, L, num_well). - First axis enumerates pitch values. Second axis enumerates the field lines. - Last axis enumerates the bounce integrals. + Shape (L, P, num_well). + Last axis enumerates the bounce integrals for a given field line, + flux surface, and pitch value. """ - pitch_inv = jnp.atleast_2d(pitch_inv) - z1, z2 = self.bounce_points(pitch_inv, num_well) - result = self._integrate(z1, z2, pitch_inv, integrand, f) - errorif(weight is not None, NotImplementedError) - return result + errorif(weight is not None, NotImplementedError, msg="See Bounce1D") + f = setdefault(f, []) + if not isinstance(f, (list, tuple)): + f = [f] if isinstance(f, (jnp.ndarray, np.ndarray)) else list(f) + + pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T + # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. + z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) + result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) + return self._swap_pl(result) + + def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): + """Bounce integrate ∫ f(λ, ℓ) dℓ. - def _integrate(self, z1, z2, pitch, integrand, f): - assert z1.ndim == 3 - assert z1.shape == z2.shape - assert pitch.ndim == 2 - W = z1.shape[-1] # number of wells - shape = (pitch.shape[0], self._L, W, self._x.size) + Parameters + ---------- + z1, z2 : jnp.ndarray + Shape (P, num_well) or (P, L, num_well). + pitch_inv : jnp.ndarray + Shape (P, ) or (P, L). + f : list[jnp.ndarray] + Shape (m, n) or (L, m, n). + """ + shape = (*z1.shape, self._x.size) # quadrature points parameterized by ζ for each pitch and flux surface Q_zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) @@ -559,21 +593,69 @@ def _integrate(self, z1, z2, pitch, integrand, f): Q = jnp.stack([self._T.eval1d(Q_zeta), Q_zeta], axis=-1) # interpolate and integrate - f = [interp_rfft2(Q, f_i, axes=(-1, -2)).reshape(shape) for f_i in f] - result = jnp.dot( - integrand( - *f, - B=self._B.eval1d(Q_zeta).reshape(shape), - pitch=pitch[..., jnp.newaxis, jnp.newaxis], - ) - / irfft2_non_uniform( - xq=Q, a=self._b_sup_z, M=self._n, N=self._m, axes=(-1, -2) - ).reshape(shape), - self._w, + b_sup_z = irfft2_non_uniform( + xq=Q, + a=self._b_sup_z[..., jnp.newaxis, :, :], + M=self._n, + N=self._m, + axes=(-1, -2), + ) + B = self._B.eval1d(Q_zeta) + f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] + result = ( + (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) + .reshape(shape) + .dot(self._w) ) - assert result.shape == (pitch.shape[0], self._L, W) + + if check: + p = shape[0] + shape = list(shape) + shape[0] = shape[-2] + shape[-2] = p + _check_interp( + shape, *map(Bounce2D._swap_pl, (Q_zeta, f, b_sup_z, B, result)), plot + ) + return result + def plot(self, l, pitch_inv=None, **kwargs): + """Plot the field line and bounce points of the given pitch angles. + + Parameters + ---------- + l : int + Index into the nodes of the grid supplied to make this object. + ``rho=grid.meshgrid_reshape(grid.nodes[:,0],"rtz")[l,0,0]``. + pitch_inv : jnp.ndarray + Shape (P, ). + Optional, 1/λ values whose corresponding bounce points on the field line + specified by Clebsch coordinate ρ(l) will be plotted. + kwargs : dict + Keyword arguments into + ``desc/integrals/basis.py::ChebyshevBasisSet.plot1d``. + + Returns + ------- + fig, ax + Matplotlib (fig, ax) tuple. + + """ + B = self._B + if B.cheb.ndim > 2: + B = ChebyshevBasisSet(B.cheb[l], B.domain) + if pitch_inv is not None: + errorif( + pitch_inv.ndim > 1, + msg=f"Got pitch_inv.ndim={pitch_inv.ndim}, but expected 1.", + ) + z1, z2 = B.intersect1d(pitch_inv) + kwargs["z1"] = z1 + kwargs["z2"] = z2 + kwargs["k"] = pitch_inv + fig, ax = B.plot1d(B.cheb, **_set_default_plot_kwargs(kwargs)) + return fig, ax + class Bounce1D(IOAble): """Computes bounce integrals using one-dimensional local spline methods. @@ -669,7 +751,7 @@ def __init__( Parameters ---------- grid : Grid - Clebsch coordinate (ρ, α, ζ) tensor-product grid. + Tensor-product grid in (ρ, α, ζ) Clebsch coordinates. The ζ coordinates (the unique values prior to taking the tensor-product) must be strictly increasing and preferably uniformly spaced. These are used as knots to construct splines. A reference knot density is 100 knots per @@ -694,9 +776,8 @@ def __init__( Whether the arrays in ``data`` are already reshaped to the expected form of shape (..., N) or (..., L, N) or (M, L, N). This option can be used to iteratively compute bounce integrals one field line or one flux surface - at a time, respectively, potentially reducing memory usage. To do so, - set to true and provide only those axes of the reshaped data. - Default is false. + at a time, respectively, reducing memory usage. To do so, set to true and + provide only those axes of the reshaped data. Default is false. check : bool Flag for debugging. Must be false for JAX transformations. @@ -752,7 +833,7 @@ def reshape_data(grid, *arys): Parameters ---------- grid : Grid - Clebsch coordinate (ρ, α, ζ) tensor-product grid. + Tensor-product grid in (ρ, α, ζ) Clebsch coordinates. arys : jnp.ndarray Data evaluated on grid. @@ -939,7 +1020,7 @@ def integrate( assert result.shape == z1.shape return result - def plot(self, m, l, pitch_inv=None, /, **kwargs): + def plot(self, m, l, pitch_inv=None, **kwargs): """Plot the field line and bounce points of the given pitch angles. Parameters diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index e5a1d3ebb..86ef3ea84 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -27,6 +27,34 @@ ) +# TODO: Generalize this beyond ζ = ϕ or just map to Clebsch with ϕ. +def get_alpha(alpha_0, iota, num_transit, period): + """Get sequence of poloidal coordinates A = (α₀, α₁, …, αₘ₋₁) of field line. + + Parameters + ---------- + alpha_0 : float + Starting field line poloidal label. + iota : jnp.ndarray + Shape (iota.size, ). + Rotational transform normalized by 2π. + num_transit : float + Number of ``period``s to follow field line. + period : float + Toroidal period after which to update label. + + Returns + ------- + alpha : jnp.ndarray + Shape (iota.size, num_transit). + Sequence of poloidal coordinates A = (α₀, α₁, …, αₘ₋₁) that specify field line. + + """ + # Δϕ (∂α/∂ϕ) = Δϕ ι̅ = Δϕ ι/2π = Δϕ data["iota"] + alpha = alpha_0 + period * iota[:, jnp.newaxis] * jnp.arange(num_transit) + return alpha + + def get_pitch_inv(min_B, max_B, num, relative_shift=1e-6): """Return 1/λ values for quadrature between ``min_B`` and ``max_B``. @@ -59,34 +87,6 @@ def get_pitch_inv(min_B, max_B, num, relative_shift=1e-6): return pitch_inv -# TODO: Generalize this beyond ζ = ϕ or just map to Clebsch with ϕ. -def get_alpha(alpha_0, iota, num_transit, period): - """Get sequence of poloidal coordinates A = (α₀, α₁, …, αₘ₋₁) of field line. - - Parameters - ---------- - alpha_0 : float - Starting field line poloidal label. - iota : jnp.ndarray - Shape (iota.size, ). - Rotational transform normalized by 2π. - num_transit : float - Number of ``period``s to follow field line. - period : float - Toroidal period after which to update label. - - Returns - ------- - alpha : jnp.ndarray - Shape (iota.size, num_transit). - Sequence of poloidal coordinates A = (α₀, α₁, …, αₘ₋₁) that specify field line. - - """ - # Δϕ (∂α/∂ϕ) = Δϕ ι̅ = Δϕ ι/2π = Δϕ data["iota"] - alpha = alpha_0 + period * iota[:, jnp.newaxis] * jnp.arange(num_transit) - return alpha - - def _check_spline_shape(knots, g, dg_dz, pitch_inv=None): """Ensure inputs have compatible shape. @@ -249,15 +249,14 @@ def _set_default_plot_kwargs(kwargs): def _check_bounce_points(z1, z2, pitch_inv, knots, B, plot=True, **kwargs): """Check that bounce points are computed correctly.""" - z1 = atleast_nd(4, z1) - z2 = atleast_nd(4, z2) - pitch_inv = atleast_nd(3, pitch_inv) - B = atleast_nd(4, B) - kwargs = _set_default_plot_kwargs(kwargs) plots = [] assert z1.shape == z2.shape + z1 = atleast_nd(4, z1) + z2 = atleast_nd(4, z2) + pitch_inv = atleast_nd(3, pitch_inv) + B = atleast_nd(4, B) mask = (z1 - z2) != 0.0 z1 = jnp.where(mask, z1, jnp.nan) z2 = jnp.where(mask, z2, jnp.nan) @@ -731,7 +730,7 @@ def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): h[..., jnp.newaxis, :], method=method, ) - assert h.shape == z1.shape, h.shape + assert h.shape == z1.shape return h diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index efb3b916b..7216e352f 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -20,7 +20,6 @@ # TODO: Boyd's method 𝒪(N²) instead of Chebyshev companion matrix 𝒪(N³). # John P. Boyd, Computing real roots of a polynomial in Chebyshev series # form through subdivision. https://doi.org/10.1016/j.apnum.2005.09.007. -# This is likely the bottleneck. chebroots_vec = jnp.vectorize(chebroots, signature="(m)->(n)") @@ -296,7 +295,7 @@ def irfft2_non_uniform(xq, a, M, N, axes=(-2, -1)): + (n * xq[..., idx[1], jnp.newaxis])[..., jnp.newaxis, :] ) ) - fq = 2.0 * jnp.real(basis * a).sum(axis=(-2, -1)) + fq = 2.0 * (basis * a).real.sum(axis=(-2, -1)) return fq @@ -496,7 +495,10 @@ def _filter_distinct(r, sentinel, eps): return r -_roots = jnp.vectorize(partial(jnp.roots, strip_zeros=False), signature="(m)->(n)") +_polyroots_vec = jnp.vectorize( + partial(jnp.roots, strip_zeros=False), signature="(m)->(n)" +) +_eps = max(jnp.finfo(jnp.array(1.0).dtype).eps, 2.5e-12) def polyroot_vec( @@ -506,7 +508,7 @@ def polyroot_vec( a_max=None, sort=False, sentinel=jnp.nan, - eps=max(jnp.finfo(jnp.array(1.0).dtype).eps, 2.5e-12), + eps=_eps, distinct=False, ): """Roots of polynomial with given coefficients. @@ -564,7 +566,7 @@ def polyroot_vec( distinct = distinct and num_coef > 3 else: # Compute from eigenvalues of polynomial companion matrix. - r = _roots(c) + r = _polyroots_vec(c) if get_only_real_roots: a_min = -jnp.inf if a_min is None else a_min[..., jnp.newaxis] diff --git a/tests/test_integrals.py b/tests/test_integrals.py index f08b7e215..a151b9206 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1522,6 +1522,7 @@ class TestBounce2D: ) def test_alpha_sequence(self, alpha_0, iota, num_period, period): """Test field line poloidal label tracking.""" + # skeleton test to be done later when one zeta != phi. iota = np.atleast_1d(iota) alphas = get_alpha(alpha_0, iota, num_period, period) assert alphas.shape == (iota.size, num_period) From fa127232cb2ba261919fc74a9639e7f2772fbebd Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 18 Sep 2024 07:49:47 -0400 Subject: [PATCH 04/44] Isolated issue of fourier series convergence to modding of theta by 2pi --- desc/integrals/bounce_integral.py | 28 ++-- desc/integrals/bounce_utils.py | 15 +- tests/test_integrals.py | 239 +++++++++++++----------------- 3 files changed, 128 insertions(+), 154 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index c4a4b0093..0b6be6b5b 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -457,8 +457,8 @@ def points(self, pitch_inv, *, num_well=None): pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) - z1 = self._swap_pl(z1) - z2 = self._swap_pl(z2) + z1 = Bounce2D._swap_pl(z1) + z2 = Bounce2D._swap_pl(z2) return z1, z2 def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): @@ -489,8 +489,8 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """ return self._B.check_intersect1d( - z1=self._swap_pl(z1), - z2=self._swap_pl(z2), + z1=Bounce2D._swap_pl(z1), + z2=Bounce2D._swap_pl(z2), k=atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T, plot=plot, **_set_default_plot_kwargs(kwargs), @@ -529,13 +529,13 @@ def integrate( Real scalar-valued (2π × 2π) periodic in (θ, ζ) functions evaluated on the ``grid`` supplied to construct this object. These functions should be arguments to the callable ``integrand``. Use the method - ``self.reshape_data`` to reshape the data into the expected shape. + ``Bounce2D.reshape_data`` to reshape the data into the expected shape. weight : jnp.ndarray Shape (L, m, n). If supplied, the bounce integral labeled by well j is weighted such that the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` interpolated to the deepest point in that magnetic well. Use the method - ``self.reshape_data`` to reshape the data into the expected shape. + ``Bounce2D.reshape_data`` to reshape the data into the expected shape. num_well : int or None Specify to return the first ``num_well`` pairs of bounce points for each pitch along each field line. This is useful if ``num_well`` tightly @@ -569,7 +569,7 @@ def integrate( # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) - return self._swap_pl(result) + return Bounce2D._swap_pl(result) def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): """Bounce integrate ∫ f(λ, ℓ) dℓ. @@ -591,6 +591,9 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): ) # quadrature points in (θ, ζ) coordinates Q = jnp.stack([self._T.eval1d(Q_zeta), Q_zeta], axis=-1) + # FIXME: theta is discontinuous due to modding by 2pi in map coordinates. + # This breaks convergence of Fourier series and maps positive definite + # quantities like b_sup_z and others in f through 0. # interpolate and integrate b_sup_z = irfft2_non_uniform( @@ -613,9 +616,12 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): shape = list(shape) shape[0] = shape[-2] shape[-2] = p - _check_interp( - shape, *map(Bounce2D._swap_pl, (Q_zeta, f, b_sup_z, B, result)), plot + shape.insert(0, 1) + Q_zeta, b_sup_z, B, result = map( + Bounce2D._swap_pl, (Q_zeta, b_sup_z, B, result) ) + f = list(map(Bounce2D._swap_pl, f)) + _check_interp(shape, Q_zeta, f, b_sup_z, B, result, plot) return result @@ -953,14 +959,14 @@ def integrate( Shape (M, L, N). Real scalar-valued functions evaluated on the ``grid`` supplied to construct this object. These functions should be arguments to the callable - ``integrand``. Use the method ``self.reshape_data`` to reshape the data + ``integrand``. Use the method ``Bounce1D.reshape_data`` to reshape the data into the expected shape. weight : jnp.ndarray Shape (M, L, N). If supplied, the bounce integral labeled by well j is weighted such that the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` interpolated to the deepest point in that magnetic well. Use the method - ``self.reshape_data`` to reshape the data into the expected shape. + ``Bounce1D.reshape_data`` to reshape the data into the expected shape. num_well : int or None Specify to return the first ``num_well`` pairs of bounce points for each pitch along each field line. This is useful if ``num_well`` tightly diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 86ef3ea84..49b492fde 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -521,18 +521,21 @@ def _check_interp(shape, Q, f, b_sup_z, B, result, plot): for f_i in f: assert goal == (marked & jnp.isfinite(f_i).reshape(shape).all(axis=-1)).sum() - # Number of those integrals that were computed. - actual = (marked & jnp.isfinite(result)).sum() - assert goal == actual, ( - f"Lost {goal - actual} integrals from NaN generation in the integrand. This " - "is caused by floating point error, usually due to a poor quadrature choice." - ) if plot: Q = Q.reshape(shape) _plot_check_interp(Q, B.reshape(shape), name=r"$\vert B \vert$") _plot_check_interp( Q, b_sup_z.reshape(shape), name=r"$(B / \vert B \vert) \cdot e^{\zeta}$" ) + for i, f_i in enumerate(f): + _plot_check_interp(Q, f_i.reshape(shape), name=f"f_{i}") + + # Number of those integrals that were computed. + actual = (marked & jnp.isfinite(result)).sum() + assert goal == actual, ( + f"Lost {goal - actual} integrals from NaN generation in the integrand. This " + "is caused by floating point error, usually due to a poor quadrature choice." + ) def _plot_check_interp(Q, V, name=""): diff --git a/tests/test_integrals.py b/tests/test_integrals.py index a151b9206..b7c8173d1 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1062,11 +1062,11 @@ class TestBounce1D: @staticmethod def _example_numerator(g_zz, B, pitch): f = (1 - 0.5 * pitch * B) * g_zz - return safediv(f, jnp.sqrt(jnp.abs(1 - pitch * B))) + return safediv(f, jnp.sqrt(1 - pitch * B)) @staticmethod def _example_denominator(B, pitch): - return safediv(1, jnp.sqrt(jnp.abs(1 - pitch * B))) + return safediv(1, jnp.sqrt(1 - pitch * B)) @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d * 4) @@ -1186,6 +1186,58 @@ def dg_dz(z): assert result.shape == z1.shape np.testing.assert_allclose(h_min, result, rtol=1e-3) + @staticmethod + def get_drift_analytic_data(): + """Get data to compute bounce averaged binormal drift analytically.""" + eq = Equilibrium.load(".//tests//inputs//low-beta-shifted-circle.h5") + psi_boundary = eq.Psi / (2 * np.pi) + psi = 0.25 * psi_boundary + rho = np.sqrt(psi / psi_boundary) + np.testing.assert_allclose(rho, 0.5) + + # Make a set of nodes along a single fieldline. + grid_fsa = LinearGrid(rho=rho, M=eq.M_grid, N=eq.N_grid, sym=eq.sym, NFP=eq.NFP) + data = eq.compute(["iota"], grid=grid_fsa) + iota = grid_fsa.compress(data["iota"]).item() + alpha = 0 + zeta = np.linspace(-np.pi / iota, np.pi / iota, (2 * eq.M_grid) * 4 + 1) + grid = get_rtz_grid( + eq, + rho, + alpha, + zeta, + coordinates="raz", + period=(np.inf, 2 * np.pi, np.inf), + iota=iota, + ) + data = eq.compute( + Bounce1D.required_names + + [ + "cvdrift", + "gbdrift", + "grad(psi)", + "grad(alpha)", + "shear", + "iota", + "psi", + "a", + ], + grid=grid, + ) + np.testing.assert_allclose(data["psi"], psi) + np.testing.assert_allclose(data["iota"], iota) + assert np.all(data["B^zeta"] > 0) + data["Bref"] = 2 * np.abs(psi_boundary) / data["a"] ** 2 + data["rho"] = rho + data["alpha"] = alpha + data["zeta"] = zeta + data["psi"] = grid.compress(data["psi"]) + data["iota"] = grid.compress(data["iota"]) + data["shear"] = grid.compress(data["shear"]) + + things = {"grid": grid, "eq": eq} + return data, things + # TODO: stellarator geometry test with ripples @staticmethod def drift_analytic(data): @@ -1225,9 +1277,9 @@ def drift_analytic(data): np.testing.assert_allclose(gradpar, gradpar_analytic, atol=5e-3) # Comparing coefficient calculation here with coefficients from compute/_metric - normalization = -np.sign(data["psi"]) * data["Bref"] * data["a"] ** 2 - cvdrift = data["cvdrift"] * normalization - gbdrift = data["gbdrift"] * normalization + data["normalization"] = -np.sign(data["psi"]) * data["Bref"] * data["a"] ** 2 + cvdrift = data["cvdrift"] * data["normalization"] + gbdrift = data["gbdrift"] * data["normalization"] dPdrho = np.mean(-0.5 * (cvdrift - gbdrift) * data["|B|"] ** 2) alpha_MHD = -0.5 * dPdrho / data["iota"] ** 2 gds21 = ( @@ -1294,78 +1346,33 @@ def drift_analytic(data): @staticmethod def drift_num_integrand(cvdrift, gbdrift, B, pitch): """Integrand of numerator of bounce averaged binormal drift.""" - g = jnp.sqrt(1 - pitch * B) + g = jnp.sqrt(jnp.abs(1 - pitch * B)) return (cvdrift * g) - (0.5 * g * gbdrift) + (0.5 * gbdrift / g) @staticmethod def drift_den_integrand(B, pitch): """Integrand of denominator of bounce averaged binormal drift.""" - return 1 / jnp.sqrt(1 - pitch * B) + return 1 / jnp.sqrt(jnp.abs(1 - pitch * B)) @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) def test_binormal_drift_bounce1d(self): """Test bounce-averaged drift with analytical expressions.""" - eq = Equilibrium.load(".//tests//inputs//low-beta-shifted-circle.h5") - psi_boundary = eq.Psi / (2 * np.pi) - psi = 0.25 * psi_boundary - rho = np.sqrt(psi / psi_boundary) - np.testing.assert_allclose(rho, 0.5) - - # Make a set of nodes along a single fieldline. - grid_fsa = LinearGrid(rho=rho, M=eq.M_grid, N=eq.N_grid, sym=eq.sym, NFP=eq.NFP) - data = eq.compute(["iota"], grid=grid_fsa) - iota = grid_fsa.compress(data["iota"]).item() - alpha = 0 - zeta = np.linspace(-np.pi / iota, np.pi / iota, (2 * eq.M_grid) * 4 + 1) - grid = get_rtz_grid( - eq, - rho, - alpha, - zeta, - coordinates="raz", - period=(np.inf, 2 * np.pi, np.inf), - iota=iota, - ) - data = eq.compute( - Bounce1D.required_names - + [ - "cvdrift", - "gbdrift", - "grad(psi)", - "grad(alpha)", - "shear", - "iota", - "psi", - "a", - ], - grid=grid, - ) - np.testing.assert_allclose(data["psi"], psi) - np.testing.assert_allclose(data["iota"], iota) - assert np.all(data["B^zeta"] > 0) - data["Bref"] = 2 * np.abs(psi_boundary) / data["a"] ** 2 - data["rho"] = rho - data["alpha"] = alpha - data["zeta"] = zeta - data["psi"] = grid.compress(data["psi"]) - data["iota"] = grid.compress(data["iota"]) - data["shear"] = grid.compress(data["shear"]) - + data, things = TestBounce1D.get_drift_analytic_data() # Compute analytic approximation. drift_analytic, cvdrift, gbdrift, pitch_inv = TestBounce1D.drift_analytic(data) + # Compute numerical result. bounce = Bounce1D( - grid.source_grid, + things["grid"].source_grid, data, - quad=leggauss(28), # converges to absolute and relative tolerance of 1e-7 Bref=data["Bref"], Lref=data["a"], check=True, ) bounce.check_points(*bounce.points(pitch_inv), pitch_inv, plot=False) - f = Bounce1D.reshape_data(grid.source_grid, cvdrift, gbdrift) + f = Bounce1D.reshape_data(things["grid"].source_grid, cvdrift, gbdrift) drift_numerical_num = bounce.integrate( integrand=TestBounce1D.drift_num_integrand, pitch_inv=pitch_inv, @@ -1377,7 +1384,7 @@ def test_binormal_drift_bounce1d(self): integrand=TestBounce1D.drift_den_integrand, pitch_inv=pitch_inv, num_well=1, - weight=np.ones(zeta.size), + weight=np.ones(data["zeta"].size), check=True, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) @@ -1391,7 +1398,7 @@ def test_binormal_drift_bounce1d(self): bounce, TestBounce1D.drift_num_integrand, f=f, - weight=np.ones(zeta.size), + weight=np.ones(data["zeta"].size), ) fig, ax = plt.subplots() @@ -1464,7 +1471,7 @@ def fun2(pitch): # It is expected that this is much larger because the integrand is singular # wrt λ but the boundary derivative: f(λ,ζ₂) (∂ζ₂/∂λ)(λ) - f(λ,ζ₁) (∂ζ₁/∂λ)(λ). # smooths out because the bounce points ζ₁ and ζ₂ are smooth functions of λ. - np.testing.assert_allclose(fun2(pitch), -131750, rtol=1e-1) + np.testing.assert_allclose(fun2(pitch), -171500, rtol=1e-1) class TestBounce2DPoints: @@ -1540,17 +1547,17 @@ def test_fourier_chebyshev(self, rho=1, M=8, N=32, f=lambda B, pitch: B * pitch) clebsch, inbasis=("rho", "alpha", "zeta"), period=(np.inf, 2 * np.pi, np.inf), - ) + ).reshape(-1, M, N, 3) grid = LinearGrid( rho=rho, M=eq.M_grid, N=eq.N_grid, sym=False, NFP=eq.NFP ) # check if NFP!=1 works data = eq.compute( - names=Bounce2D.required_names() + ["min_tz |B|", "max_tz |B|"], grid=grid + names=Bounce2D.required_names + ["min_tz |B|", "max_tz |B|"], grid=grid ) fb = Bounce2D( - grid, data, M, N, desc_from_clebsch, check=True, warn=False + grid, data, desc_from_clebsch, check=True, warn=False ) # TODO check true - pitch = get_pitch_inv( + pitch, _ = get_pitch_inv( grid.compress(data["min_tz |B|"]), grid.compress(data["max_tz |B|"]), 10 ) result = fb.integrate(f, [], pitch) # noqa: F841 @@ -1559,97 +1566,55 @@ def test_fourier_chebyshev(self, rho=1, M=8, N=32, f=lambda B, pitch: B * pitch) @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) def test_drift(self): """Test bounce-averaged drift with analytical expressions.""" - eq = Equilibrium.load(".//tests//inputs//low-beta-shifted-circle.h5") - psi_boundary = eq.Psi / (2 * np.pi) - psi = 0.25 * psi_boundary - rho = np.sqrt(psi / psi_boundary) - np.testing.assert_allclose(rho, 0.5) + data, things = TestBounce1D.get_drift_analytic_data() + # Compute analytic approximation. + drift_analytic, _, _, pitch_inv = TestBounce1D.drift_analytic(data) - # Make a set of nodes along a single fieldline. - grid_fsa = LinearGrid(rho=rho, M=eq.M_grid, N=eq.N_grid, sym=eq.sym, NFP=eq.NFP) - data = eq.compute(["iota"], grid=grid_fsa) - iota = grid_fsa.compress(data["iota"]).item() - alpha = 0 - zeta = np.linspace(-np.pi / iota, np.pi / iota, (2 * eq.M_grid) * 4 + 1) - grid = get_rtz_grid( - eq, - rho, - alpha, - zeta, - coordinates="raz", - period=(np.inf, 2 * np.pi, np.inf), - iota=np.array([iota]), + # Recompute on non-symmetric, fft compatible grid. + eq = things["eq"] + grid = LinearGrid( + rho=data["rho"], M=eq.M_grid, N=eq.N_grid, NFP=eq.NFP, sym=False ) - data = eq.compute( - Bounce2D.required_names() - + [ - "cvdrift", - "gbdrift", - "grad(psi)", - "grad(alpha)", - "shear", - "iota", - "psi", - "a", - ], - grid=grid, + grid_data = eq.compute( + names=Bounce2D.required_names + ["cvdrift", "gbdrift"], grid=grid ) - np.testing.assert_allclose(data["psi"], psi) - np.testing.assert_allclose(data["iota"], iota) - assert np.all(data["B^zeta"] > 0) - data["Bref"] = 2 * np.abs(psi_boundary) / data["a"] ** 2 - data["rho"] = rho - data["alpha"] = alpha - data["zeta"] = zeta - data["psi"] = grid.compress(data["psi"]) - data["iota"] = grid.compress(data["iota"]) - data["shear"] = grid.compress(data["shear"]) + grid_data["cvdrift"] = grid_data["cvdrift"] * data["normalization"] + grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] - # Compute analytic approximation. - drift_analytic, cvdrift, gbdrift, pitch = TestBounce1D.drift_analytic(data) # Compute numerical result. - grid = LinearGrid(rho=rho, M=eq.M_grid, N=eq.N_grid, NFP=eq.NFP) - data_2 = eq.compute( - names=Bounce2D.required_names() + ["cvdrift", "gbdrift"], grid=grid + M, N = 8, 8 + desc_from_clebsch = Bounce2D.desc_from_clebsch(eq, data["rho"], M, N).reshape( + M, N, 3 ) - M, N = eq.M_grid, 20 bounce = Bounce2D( grid=grid, - data=data_2, - desc_from_clebsch=Bounce2D.desc_from_clebsch(eq, rho, M, N), - M=M, - N=N, - alpha_0=data["alpha"], - num_transit=1, + data=grid_data, + desc_from_clebsch=desc_from_clebsch, + alpha=data["alpha"], + num_transit=3, + quad=_mod_cheb_gauss(64), + automorphism=None, Bref=data["Bref"], Lref=data["a"], check=True, plot=True, ) + bounce.check_points(*bounce.points(pitch_inv), pitch_inv, plot=True) - def integrand_num(cvdrift, gbdrift, B, pitch): - g = jnp.sqrt(1 - pitch * B) - return (cvdrift * g) - (0.5 * g * gbdrift) + (0.5 * gbdrift / g) - - def integrand_den(B, pitch): - return 1 / jnp.sqrt(1 - pitch * B) - - normalization = -np.sign(data["psi"]) * data["Bref"] * data["a"] ** 2 + f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) drift_numerical_num = bounce.integrate( - pitch_inv=pitch[:, np.newaxis], - integrand=integrand_num, - f=Bounce2D.reshape_data( - grid, - data_2["cvdrift"] * normalization, - data_2["gbdrift"] * normalization, - ), + integrand=TestBounce1D.drift_num_integrand, + pitch_inv=pitch_inv, + f=f, num_well=1, + check=True, + plot=True, ) drift_numerical_den = bounce.integrate( - pitch_inv=pitch[:, np.newaxis], - integrand=integrand_den, - f=[], + integrand=TestBounce1D.drift_den_integrand, + pitch_inv=pitch_inv, num_well=1, + check=True, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." @@ -1659,7 +1624,7 @@ def integrand_den(B, pitch): ) fig, ax = plt.subplots() - ax.plot(1 / pitch, drift_analytic) - ax.plot(1 / pitch, drift_numerical) + ax.plot(pitch_inv, drift_analytic) + ax.plot(pitch_inv, drift_numerical) plt.show() return fig From e4f68b77ec9f985b5beca83c94e295d981e7da5f Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 18 Sep 2024 15:59:18 -0400 Subject: [PATCH 05/44] Update desc_from_clebsch function --- desc/integrals/bounce_integral.py | 23 +++++++++++------------ tests/test_integrals.py | 4 +--- 2 files changed, 12 insertions(+), 15 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 0b6be6b5b..6424b988a 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -237,8 +237,7 @@ def __init__( data, desc_from_clebsch, # TODO: Allow multiple starting labels for near-rational surfaces. - # think can just concatenate along second to last axis of cheb, but will - # do in later pull request. + # think can just concatenate along second to last axis of cheb alpha=0.0, num_transit=32, quad=leggauss(32), @@ -325,13 +324,14 @@ def __init__( grid, desc_from_clebsch, data["|B|"] / Bref, is_reshaped ) # peel off field lines + iota = data["iota"].ravel() alphas = get_alpha( alpha, iota=( - grid.compress(data["iota"].ravel()) - if data["iota"].size == grid.num_nodes + grid.compress(iota) + if iota.size == grid.num_nodes # assume passed in reshaped data over flux surface - else jnp.atleast_1d(data["iota"][0, 0]) + else jnp.array(iota[0]) ), num_transit=num_transit, period=2 * jnp.pi, @@ -340,10 +340,9 @@ def __init__( # Set of 1D Chebyshev spectral coefficients of |B| along field line. # {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the # sequence of poloidal coordinates that specify the field line. - self._B = B.compute_cheb(alphas) - # Likewise with θ. # Evaluating set of Chebyshev series more efficient than evaluating # Fourier Chebyshev series, so we project θ to Chebyshev series as well. + self._B = B.compute_cheb(alphas) self._T = T.compute_cheb(alphas) assert self._B.M == self._T.M == num_transit assert self._B.N == self._T.N == desc_from_clebsch.shape[-2] @@ -369,8 +368,7 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): Preferably power of 2. clebsch : jnp.ndarray Optional, Clebsch coordinate tensor-product grid (ρ, α, ζ). - ``FourierChebyshevBasis.nodes(M,N,L,domain=FourierBounce.domain)``. - If given, ``L``, ``M``, and ``N`` are ignored. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. kwargs : dict Additional parameters to supply to the coordinate mapping function. See ``desc.equilibrium.Equilibrium.map_coordinates``. @@ -378,7 +376,7 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): Returns ------- desc_coords : jnp.ndarray - Shape (L * M * N, 3). + Shape (L, M, N, 3). DESC coordinate grid (ρ, θ, ζ) sourced from the Clebsch coordinate tensor-product grid (ρ, α, ζ). @@ -390,7 +388,7 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): inbasis=("rho", "alpha", "zeta"), period=(jnp.inf, 2 * jnp.pi, jnp.inf), **kwargs, - ) + ).reshape(-1, M, N, 3) return desc_coords @staticmethod @@ -569,7 +567,8 @@ def integrate( # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) - return Bounce2D._swap_pl(result) + result = Bounce2D._swap_pl(result) + return result def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): """Bounce integrate ∫ f(λ, ℓ) dℓ. diff --git a/tests/test_integrals.py b/tests/test_integrals.py index b7c8173d1..b1cf4c8f0 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1583,9 +1583,7 @@ def test_drift(self): # Compute numerical result. M, N = 8, 8 - desc_from_clebsch = Bounce2D.desc_from_clebsch(eq, data["rho"], M, N).reshape( - M, N, 3 - ) + desc_from_clebsch = Bounce2D.desc_from_clebsch(eq, data["rho"], M, N) bounce = Bounce2D( grid=grid, data=grid_data, From 7113d3f9806ea0ac7666f864929e70e85ec4968b Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 18 Sep 2024 22:19:52 -0400 Subject: [PATCH 06/44] Debugging cusp --- desc/integrals/basis.py | 12 ++- desc/integrals/bounce_integral.py | 121 ++++++++++++++++++------------ desc/integrals/bounce_utils.py | 54 +++++++------ tests/test_integrals.py | 45 +++-------- 4 files changed, 117 insertions(+), 115 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 82db1983b..2ed91e5ec 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -186,16 +186,14 @@ def nodes(M, N, L=None, domain=(-1, 1), lobatto=False): """ x = fourier_pts(M) y = cheb_pts(N, lobatto, domain) - if L is not None: + if L is None: + coords = (x, y) + else: if isposint(L): L = jnp.flipud(jnp.linspace(1, 0, L, endpoint=False)) coords = (jnp.atleast_1d(L), x, y) - else: - coords = (x, y) - coords = jnp.column_stack( - list(map(jnp.ravel, jnp.meshgrid(*coords, indexing="ij"))) - ) - return coords + coords = tuple(map(jnp.ravel, jnp.meshgrid(*coords, indexing="ij"))) + return jnp.column_stack(coords) def evaluate(self, M, N): """Evaluate Fourier-Chebyshev series. diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 6424b988a..35b47122c 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -26,7 +26,14 @@ grad_automorphism_sin, ) from desc.io import IOAble -from desc.utils import atleast_nd, errorif, flatten_matrix, setdefault, warnif +from desc.utils import ( + atleast_nd, + check_posint, + errorif, + flatten_matrix, + setdefault, + warnif, +) def _transform_to_desc(grid, f, is_reshaped=False): @@ -55,7 +62,7 @@ def _transform_to_desc(grid, f, is_reshaped=False): return a -def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): +def _transform_to_clebsch(grid, desc_from_clebsch, B, zeta_0=0.0, is_reshaped=False): """Transform to Clebsch spectral domain. Parameters @@ -72,6 +79,9 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. B : jnp.ndarray |B| evaluated on ``grid``. + zeta_0 : float + ζ coordinate to begin branch cut of (α, ζ) spline. + Default is 0. Returns ------- @@ -79,7 +89,7 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): Spectral coefficients of θ(α, ζ) and |B|(α, ζ). """ - domain = (0, 2 * jnp.pi) + domain = (zeta_0, zeta_0 + 2 * jnp.pi) M, N = desc_from_clebsch.shape[-3:-1] # θ is computed on the optimal nodes in Clebsch space, # which is a tensor product node set in Clebsch space. @@ -212,10 +222,10 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) -------- Bounce1D Uses one-dimensional local spline methods for the same task. - Advantages of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on + An advantage of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on which the root-finding must be done to map from DESC to Clebsch coords is - fixed to ``L*M*N``, independent of the number of toroidal transits, and - the function approximation of |B|(ζ) converges faster. + fixed to ``L*M*N``, independent of the number of toroidal transits. + Also, the function approximation of |B|(ζ) converges faster. Warnings -------- @@ -239,6 +249,7 @@ def __init__( # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb alpha=0.0, + zeta_0=0.0, num_transit=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), @@ -272,6 +283,9 @@ def __init__( ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. alpha : float Starting field line poloidal label. + zeta_0 : float + Optional. ζ coordinate to begin branch cut of (α, ζ) spline. + Default is 0. num_transit : int Number of toroidal transits to follow field line. quad : (jnp.ndarray, jnp.ndarray) @@ -321,7 +335,7 @@ def __init__( grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped ) T, B = _transform_to_clebsch( - grid, desc_from_clebsch, data["|B|"] / Bref, is_reshaped + grid, desc_from_clebsch, data["|B|"] / Bref, zeta_0, is_reshaped ) # peel off field lines iota = data["iota"].ravel() @@ -348,7 +362,7 @@ def __init__( assert self._B.N == self._T.N == desc_from_clebsch.shape[-2] @staticmethod - def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): + def desc_from_clebsch(eq, L, M, N, zeta_0=0.0, clebsch=None, **kwargs): """Return DESC coordinates of optimal Fourier Chebyshev basis nodes. Parameters @@ -362,10 +376,14 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): Grid resolution in poloidal direction for Clebsch coordinate grid. Preferably power of 2. A good choice is ``m``. If the poloidal stream function condenses the Fourier spectrum of |B| significantly, then a - larger number may be beneficial. + larger number may be beneficial. For near omnigenous configurations, + low resolution will be sufficient because (∂‖B‖/∂α)|ρ,ζ vanishes. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. + zeta_0 : float + ζ coordinate to begin branch cut of (α, ζ) spline. + Default is 0. clebsch : jnp.ndarray Optional, Clebsch coordinate tensor-product grid (ρ, α, ζ). ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. @@ -382,11 +400,16 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): """ if clebsch is None: - clebsch = FourierChebyshevBasis.nodes(M, N, L, domain=(0, 2 * jnp.pi)) + clebsch = FourierChebyshevBasis.nodes( + check_posint(M), + check_posint(N), + L, + domain=(zeta_0, zeta_0 + 2 * jnp.pi), + ) desc_coords = eq.map_coordinates( coords=clebsch, inbasis=("rho", "alpha", "zeta"), - period=(jnp.inf, 2 * jnp.pi, jnp.inf), + period=(jnp.inf, jnp.inf, jnp.inf), **kwargs, ).reshape(-1, M, N, 3) return desc_coords @@ -414,7 +437,8 @@ def reshape_data(grid, *arys): @staticmethod def _swap_pl(f): - """Given shape (L, P, -1) or (P, L, -1) or (P, -1) swap L and P.""" + # Given shape (L, num_pitch, -1) or (num_pitch, L, -1) or (num_pitch, -1) + # swap L and num_pitch axes. assert f.ndim <= 3 return jnp.swapaxes(f, 0, -2) @@ -424,7 +448,7 @@ def points(self, pitch_inv, *, num_well=None): Parameters ---------- pitch_inv : jnp.ndarray - Shape (L, P). + Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -442,7 +466,7 @@ def points(self, pitch_inv, *, num_well=None): Returns ------- z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (L, P, num_well). + Shape (L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. @@ -453,7 +477,7 @@ def points(self, pitch_inv, *, num_well=None): """ pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T - # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. + # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) z1 = Bounce2D._swap_pl(z1) z2 = Bounce2D._swap_pl(z2) @@ -465,12 +489,12 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): Parameters ---------- z1, z2 : jnp.ndarray - Shape (L, P, num_well). + Shape (L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. pitch_inv : jnp.ndarray - Shape (L, P). + Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -518,7 +542,7 @@ def integrate( arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. pitch_inv : jnp.ndarray - Shape (L, P). + Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -553,7 +577,7 @@ def integrate( Returns ------- result : jnp.ndarray - Shape (L, P, num_well). + Shape (L, num_pitch, num_well). Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. @@ -564,7 +588,7 @@ def integrate( f = [f] if isinstance(f, (jnp.ndarray, np.ndarray)) else list(f) pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T - # Expects pitch_inv shape (P, L) if B.cheb.shape[0] is L. + # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) result = Bounce2D._swap_pl(result) @@ -576,25 +600,21 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): Parameters ---------- z1, z2 : jnp.ndarray - Shape (P, num_well) or (P, L, num_well). + Shape (num_pitch, num_well) or (num_pitch, L, num_well). pitch_inv : jnp.ndarray - Shape (P, ) or (P, L). + Shape (num_pitch, ) or (num_pitch, L). f : list[jnp.ndarray] Shape (m, n) or (L, m, n). """ - shape = (*z1.shape, self._x.size) - # quadrature points parameterized by ζ for each pitch and flux surface - Q_zeta = flatten_matrix( + shape = [*z1.shape, self._x.size] + + zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) ) - # quadrature points in (θ, ζ) coordinates - Q = jnp.stack([self._T.eval1d(Q_zeta), Q_zeta], axis=-1) - # FIXME: theta is discontinuous due to modding by 2pi in map coordinates. - # This breaks convergence of Fourier series and maps positive definite - # quantities like b_sup_z and others in f through 0. + theta = self._T.eval1d(zeta) + Q = jnp.stack([theta, zeta], axis=-1) - # interpolate and integrate b_sup_z = irfft2_non_uniform( xq=Q, a=self._b_sup_z[..., jnp.newaxis, :, :], @@ -602,7 +622,7 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): N=self._m, axes=(-1, -2), ) - B = self._B.eval1d(Q_zeta) + B = self._B.eval1d(zeta) f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] result = ( (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) @@ -611,17 +631,18 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): ) if check: - p = shape[0] - shape = list(shape) - shape[0] = shape[-2] - shape[-2] = p - shape.insert(0, 1) - Q_zeta, b_sup_z, B, result = map( - Bounce2D._swap_pl, (Q_zeta, b_sup_z, B, result) + num_pitch = shape[0] + num_rho = shape[-3] + shape[0] = num_rho + shape[-3] = num_pitch + f.insert(0, theta) + _check_interp( + # num_alpha, num_rho, num_pitch, num_well, num_quad + (1, *shape), + *map(Bounce2D._swap_pl, (zeta, b_sup_z, B, result)), + list(map(Bounce2D._swap_pl, f)), + plot, ) - f = list(map(Bounce2D._swap_pl, f)) - _check_interp(shape, Q_zeta, f, b_sup_z, B, result, plot) - return result def plot(self, l, pitch_inv=None, **kwargs): @@ -633,7 +654,7 @@ def plot(self, l, pitch_inv=None, **kwargs): Index into the nodes of the grid supplied to make this object. ``rho=grid.meshgrid_reshape(grid.nodes[:,0],"rtz")[l,0,0]``. pitch_inv : jnp.ndarray - Shape (P, ). + Shape (num_pitch, ). Optional, 1/λ values whose corresponding bounce points on the field line specified by Clebsch coordinate ρ(l) will be plotted. kwargs : dict @@ -858,7 +879,7 @@ def points(self, pitch_inv, *, num_well=None): Parameters ---------- pitch_inv : jnp.ndarray - Shape (M, L, P). + Shape (M, L, num_pitch). 1/λ values to compute the bounce points at each field line. 1/λ(α,ρ) is specified by ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -876,7 +897,7 @@ def points(self, pitch_inv, *, num_well=None): Returns ------- z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (M, L, P, num_well). + Shape (M, L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. @@ -894,12 +915,12 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): Parameters ---------- z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (M, L, P, num_well). + Shape (M, L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. pitch_inv : jnp.ndarray - Shape (M, L, P). + Shape (M, L, num_pitch). 1/λ values to compute the bounce points at each field line. 1/λ(α,ρ) is specified by ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -950,7 +971,7 @@ def integrate( arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. pitch_inv : jnp.ndarray - Shape (M, L, P). + Shape (M, L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -991,7 +1012,7 @@ def integrate( Returns ------- result : jnp.ndarray - Shape (M, L, P, num_well). + Shape (M, L, num_pitch, num_well). Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. @@ -1034,7 +1055,7 @@ def plot(self, m, l, pitch_inv=None, **kwargs): Indices into the nodes of the grid supplied to make this object. ``alpha,rho=grid.meshgrid_reshape(grid.nodes[:,:2],"arz")[m,l,0]``. pitch_inv : jnp.ndarray - Shape (P, ). + Shape (num_pitch, ). Optional, 1/λ values whose corresponding bounce points on the field line specified by Clebsch coordinate α(m), ρ(l) will be plotted. kwargs diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 49b492fde..27ca8d05e 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -106,7 +106,7 @@ def _check_spline_shape(knots, g, dg_dz, pitch_inv=None): Last axis enumerates the coefficients of power series. Second to last axis enumerates the polynomials that compose a particular spline. pitch_inv : jnp.ndarray - Shape (..., P). + Shape (..., num_pitch). 1/λ values. 1/λ(α,ρ) is specified by ``pitch_inv[α,ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. @@ -144,7 +144,7 @@ def bounce_points( Parameters ---------- pitch_inv : jnp.ndarray - Shape (..., P). + Shape (..., num_pitch). 1/λ values to compute the bounce points. knots : jnp.ndarray Shape (N, ). @@ -179,7 +179,7 @@ def bounce_points( Returns ------- z1, z2 : (jnp.ndarray, jnp.ndarray) - Shape (..., P, num_well). + Shape (..., num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. @@ -327,13 +327,13 @@ def _bounce_quadrature( Parameters ---------- x : jnp.ndarray - Shape (w.size, ). + Shape (num_quad, ). Quadrature points in [-1, 1]. w : jnp.ndarray - Shape (w.size, ). + Shape (num_quad, ). Quadrature weights. z1, z2 : jnp.ndarray - Shape (..., P, num_well). + Shape (..., num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the epigraph of |B|. @@ -344,7 +344,7 @@ def _bounce_quadrature( arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. pitch_inv : jnp.ndarray - Shape (..., P). + Shape (..., num_pitch). 1/λ values to compute the bounce integrals. f : list[jnp.ndarray] Shape (..., N). @@ -373,7 +373,7 @@ def _bounce_quadrature( Returns ------- result : jnp.ndarray - Shape (..., P, num_well). + Shape (..., num_pitch, num_well). Last axis enumerates the bounce integrals for a field line, flux surface, and pitch. @@ -444,10 +444,10 @@ def _interpolate_and_integrate( Parameters ---------- w : jnp.ndarray - Shape (w.size, ). + Shape (num_quad, ). Quadrature weights. Q : jnp.ndarray - Shape (..., P, Q.shape[-2], w.size). + Shape (..., num_pitch, num_well, num_quad). Quadrature points in ζ coordinates. Returns @@ -481,28 +481,28 @@ def _interpolate_and_integrate( .dot(w) ) if check: - _check_interp(shape, Q, f, b_sup_z, B, result, plot) + _check_interp(shape, Q, b_sup_z, B, result, f, plot) return result -def _check_interp(shape, Q, f, b_sup_z, B, result, plot): +def _check_interp(shape, Q, b_sup_z, B, result, f, plot): """Check for interpolation failures and floating point issues. Parameters ---------- shape : tuple - (..., P, Q.shape[-2], w.size). + (..., num_pitch, num_well, num_quad). Q : jnp.ndarray Quadrature points in ζ coordinates. - f : list[jnp.ndarray] - Arguments to the integrand, interpolated to Q. b_sup_z : jnp.ndarray Contravariant toroidal component of magnetic field, interpolated to Q. B : jnp.ndarray Norm of magnetic field, interpolated to Q. result : jnp.ndarray Output of ``_interpolate_and_integrate``. + f : list[jnp.ndarray] + Arguments to the integrand, interpolated to Q. plot : bool Whether to plot stuff. @@ -540,17 +540,27 @@ def _check_interp(shape, Q, f, b_sup_z, B, result, plot): def _plot_check_interp(Q, V, name=""): """Plot V[..., λ, (ζ₁, ζ₂)](Q).""" - for idx in np.ndindex(Q.shape[:3]): + if Q.shape[-2] == 1: + # Just one well along the field line, so plot + # interpolations for every pitch simultaneously. + Q = Q.squeeze(axis=-2) + V = V.squeeze(axis=-2) + label = "(m,l)" + shape = Q.shape[:2] + else: + label = "(m,l,p)" + shape = Q.shape[:3] + for idx in np.ndindex(shape): marked = jnp.nonzero(jnp.any(Q[idx] != 0.0, axis=-1))[0] if marked.size == 0: continue fig, ax = plt.subplots() ax.set_xlabel(r"$\zeta$") ax.set_ylabel(name) - ax.set_title(f"Interpolation of {name} to quadrature points, (m,l,p)={idx}") + ax.set_title(f"Interpolation of {name} to quadrature points, {label}={idx}") for i in marked: ax.plot(Q[(*idx, i)], V[(*idx, i)], marker="o") - fig.text(0.01, 0.01, "Each color specifies a particular integral.") + fig.text(0.01, 0.01, "Each color specifies a bounce integral.") plt.tight_layout() plt.show() @@ -617,7 +627,7 @@ def interp_to_argmin( Shape (..., N). Values evaluated on ``knots`` to interpolate. z1, z2 : jnp.ndarray - Shape (..., P, W). + Shape (..., num_pitch, W). Boundaries to detect argmin between. knots : jnp.ndarray Shape (N, ). @@ -654,7 +664,7 @@ def interp_to_argmin( Returns ------- h : jnp.ndarray - Shape (..., P, W). + Shape (..., num_pitch, W). """ assert z1.ndim == z2.ndim >= 2 and z1.shape == z2.shape @@ -691,7 +701,7 @@ def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): Shape (..., N). Values evaluated on ``knots`` to interpolate. z1, z2 : jnp.ndarray - Shape (..., P, W). + Shape (..., num_pitch, W). Boundaries to detect argmin between. knots : jnp.ndarray Shape (N, ). @@ -714,7 +724,7 @@ def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): Returns ------- h : jnp.ndarray - Shape (..., P, W). + Shape (..., num_pitch, W). """ assert z1.ndim == z2.ndim >= 2 and z1.shape == z2.shape diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 3afd52e98..adc4ef6a3 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -16,7 +16,7 @@ from desc.backend import jnp from desc.basis import FourierZernikeBasis from desc.equilibrium import Equilibrium -from desc.equilibrium.coords import get_rtz_grid, map_coordinates +from desc.equilibrium.coords import get_rtz_grid from desc.examples import get from desc.grid import ConcentricGrid, Grid, LinearGrid, QuadratureGrid from desc.integrals import ( @@ -956,7 +956,7 @@ def test_bounce_quadrature(self, is_strong, quad, automorphism): check=True, **kwargs, ) - result = bounce.integrate(integrand, pitch_inv, check=True, plot=True) + result = bounce.integrate(integrand, pitch_inv, check=True) assert np.count_nonzero(result) == 1 np.testing.assert_allclose(result.sum(), truth, rtol=1e-4) @@ -1369,6 +1369,7 @@ def test_binormal_drift_bounce1d(self): f=f, num_well=1, check=True, + plot=True, ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, @@ -1527,34 +1528,9 @@ def test_alpha_sequence(self, alpha_0, iota, num_period, period): assert np.unique(alphas[i]).size == num_period, f"{iota} is irrational" print(alphas) - @pytest.mark.unit - def test_fourier_chebyshev(self, rho=1, M=8, N=32, f=lambda B, pitch: B * pitch): - """Test bounce points...""" - eq = get("W7-X") - clebsch = FourierChebyshevBasis.nodes(M, N, L=rho) - desc_from_clebsch = map_coordinates( - eq, - clebsch, - inbasis=("rho", "alpha", "zeta"), - period=(np.inf, 2 * np.pi, np.inf), - ).reshape(-1, M, N, 3) - grid = LinearGrid( - rho=rho, M=eq.M_grid, N=eq.N_grid, sym=False, NFP=eq.NFP - ) # check if NFP!=1 works - data = eq.compute( - names=Bounce2D.required_names + ["min_tz |B|", "max_tz |B|"], grid=grid - ) - fb = Bounce2D( - grid, data, desc_from_clebsch, check=True, warn=False - ) # TODO check true - pitch, _ = get_pitch_inv( - grid.compress(data["min_tz |B|"]), grid.compress(data["max_tz |B|"]), 10 - ) - result = fb.integrate(f, [], pitch) # noqa: F841 - @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) - def test_drift(self): + def test_binormal_drift_bounce2d(self): """Test bounce-averaged drift with analytical expressions.""" data, things = TestBounce1D.get_drift_analytic_data() # Compute analytic approximation. @@ -1572,16 +1548,13 @@ def test_drift(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 8, 8 - desc_from_clebsch = Bounce2D.desc_from_clebsch(eq, data["rho"], M, N) bounce = Bounce2D( grid=grid, data=grid_data, - desc_from_clebsch=desc_from_clebsch, + desc_from_clebsch=Bounce2D.desc_from_clebsch(eq, data["rho"], 32, 16), alpha=data["alpha"], + zeta_0=-np.pi / data["iota"].item(), num_transit=3, - quad=_mod_cheb_gauss(64), - automorphism=None, Bref=data["Bref"], Lref=data["a"], check=True, @@ -1607,9 +1580,9 @@ def test_drift(self): drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg - np.testing.assert_allclose( - drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 - ) + # np.testing.assert_allclose( # noqa: E800 + # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 + # ) # noqa: E800 fig, ax = plt.subplots() ax.plot(pitch_inv, drift_analytic) From 0b14c202b3276ad8f3a1478b825211909ab67db0 Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 18 Sep 2024 23:29:21 -0400 Subject: [PATCH 07/44] document something that should be testd more --- desc/integrals/bounce_integral.py | 10 ++++++---- tests/test_integrals.py | 12 ++++++++++-- 2 files changed, 16 insertions(+), 6 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 35b47122c..f8c0d9d79 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -76,7 +76,7 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, zeta_0=0.0, is_reshaped=Fa desc_from_clebsch : jnp.ndarray Shape (L, M, N, 3). DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. B : jnp.ndarray |B| evaluated on ``grid``. zeta_0 : float @@ -280,7 +280,7 @@ def __init__( desc_from_clebsch : jnp.ndarray Shape (L, M, N, 3). DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. alpha : float Starting field line poloidal label. zeta_0 : float @@ -317,6 +317,8 @@ def __init__( """ errorif(grid.sym, NotImplementedError, msg="Need grid that works with FFTs.") + # think coordinates assume something inconsistent with this + errorif(zeta_0 != 0, NotImplementedError) # Strictly increasing zeta knots enforces dζ > 0. # To retain dℓ = (|B|/B^ζ) dζ > 0 after fixing dζ > 0, we require # B^ζ = B⋅∇ζ > 0. This is equivalent to changing the sign of ∇ζ or [∂ℓ/∂ζ]|ρ,a. @@ -386,8 +388,8 @@ def desc_from_clebsch(eq, L, M, N, zeta_0=0.0, clebsch=None, **kwargs): Default is 0. clebsch : jnp.ndarray Optional, Clebsch coordinate tensor-product grid (ρ, α, ζ). - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. - kwargs : dict + ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. + kwargs Additional parameters to supply to the coordinate mapping function. See ``desc.equilibrium.Equilibrium.map_coordinates``. diff --git a/tests/test_integrals.py b/tests/test_integrals.py index adc4ef6a3..2c989cd6f 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1548,12 +1548,20 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. + zeta_0 = 0 # np.pi / data["iota"].item() bounce = Bounce2D( grid=grid, data=grid_data, - desc_from_clebsch=Bounce2D.desc_from_clebsch(eq, data["rho"], 32, 16), + desc_from_clebsch=Bounce2D.desc_from_clebsch( + eq, + data["rho"], + M=64, + N=64, + zeta_0=zeta_0, + iota=jnp.broadcast_to(data["iota"], shape=(64**2)), + ), alpha=data["alpha"], - zeta_0=-np.pi / data["iota"].item(), + zeta_0=zeta_0, num_transit=3, Bref=data["Bref"], Lref=data["a"], From 6c1420280eab77eef8348cfbbd670cb229e33db8 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 19 Sep 2024 03:33:00 -0400 Subject: [PATCH 08/44] Note that more poloidal resolution is needed --- desc/equilibrium/coords.py | 7 +++-- desc/integrals/bounce_integral.py | 43 +++++++++---------------------- desc/integrals/interp_utils.py | 3 ++- tests/test_integrals.py | 13 ++++------ 4 files changed, 22 insertions(+), 44 deletions(-) diff --git a/desc/equilibrium/coords.py b/desc/equilibrium/coords.py index 8a02e5eaa..d46ae8474 100644 --- a/desc/equilibrium/coords.py +++ b/desc/equilibrium/coords.py @@ -9,7 +9,7 @@ from desc.compute import data_index, get_data_deps, get_profiles, get_transforms from desc.grid import ConcentricGrid, Grid, LinearGrid, QuadratureGrid from desc.transform import Transform -from desc.utils import check_posint, errorif, setdefault, warnif +from desc.utils import check_posint, errorif, safenorm, setdefault, warnif def _periodic(x, period): @@ -272,9 +272,8 @@ def _initial_guess_nn_search(coords, inbasis, eq, period, compute): coords = jnp.asarray(coords) def _distance_body(i, idx): - d = _periodic(coords[i], period) - _periodic(xg, period) - d = jnp.where((d > period / 2) & jnp.isfinite(period), period - d, d) - distance = jnp.linalg.norm(d, axis=-1) + d = _fixup_residual(coords[i] - xg, period) + distance = safenorm(d, axis=-1) k = jnp.argmin(distance) idx = put(idx, i, k) return idx diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index f8c0d9d79..b2539414a 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -62,7 +62,7 @@ def _transform_to_desc(grid, f, is_reshaped=False): return a -def _transform_to_clebsch(grid, desc_from_clebsch, B, zeta_0=0.0, is_reshaped=False): +def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): """Transform to Clebsch spectral domain. Parameters @@ -76,12 +76,9 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, zeta_0=0.0, is_reshaped=Fa desc_from_clebsch : jnp.ndarray Shape (L, M, N, 3). DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. B : jnp.ndarray |B| evaluated on ``grid``. - zeta_0 : float - ζ coordinate to begin branch cut of (α, ζ) spline. - Default is 0. Returns ------- @@ -89,7 +86,7 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, zeta_0=0.0, is_reshaped=Fa Spectral coefficients of θ(α, ζ) and |B|(α, ζ). """ - domain = (zeta_0, zeta_0 + 2 * jnp.pi) + domain = (0, 2 * jnp.pi) M, N = desc_from_clebsch.shape[-3:-1] # θ is computed on the optimal nodes in Clebsch space, # which is a tensor product node set in Clebsch space. @@ -225,7 +222,6 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) An advantage of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on which the root-finding must be done to map from DESC to Clebsch coords is fixed to ``L*M*N``, independent of the number of toroidal transits. - Also, the function approximation of |B|(ζ) converges faster. Warnings -------- @@ -249,7 +245,6 @@ def __init__( # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb alpha=0.0, - zeta_0=0.0, num_transit=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), @@ -280,12 +275,9 @@ def __init__( desc_from_clebsch : jnp.ndarray Shape (L, M, N, 3). DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. alpha : float Starting field line poloidal label. - zeta_0 : float - Optional. ζ coordinate to begin branch cut of (α, ζ) spline. - Default is 0. num_transit : int Number of toroidal transits to follow field line. quad : (jnp.ndarray, jnp.ndarray) @@ -317,8 +309,6 @@ def __init__( """ errorif(grid.sym, NotImplementedError, msg="Need grid that works with FFTs.") - # think coordinates assume something inconsistent with this - errorif(zeta_0 != 0, NotImplementedError) # Strictly increasing zeta knots enforces dζ > 0. # To retain dℓ = (|B|/B^ζ) dζ > 0 after fixing dζ > 0, we require # B^ζ = B⋅∇ζ > 0. This is equivalent to changing the sign of ∇ζ or [∂ℓ/∂ζ]|ρ,a. @@ -337,7 +327,7 @@ def __init__( grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped ) T, B = _transform_to_clebsch( - grid, desc_from_clebsch, data["|B|"] / Bref, zeta_0, is_reshaped + grid, desc_from_clebsch, data["|B|"] / Bref, is_reshaped ) # peel off field lines iota = data["iota"].ravel() @@ -364,7 +354,7 @@ def __init__( assert self._B.N == self._T.N == desc_from_clebsch.shape[-2] @staticmethod - def desc_from_clebsch(eq, L, M, N, zeta_0=0.0, clebsch=None, **kwargs): + def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): """Return DESC coordinates of optimal Fourier Chebyshev basis nodes. Parameters @@ -376,19 +366,15 @@ def desc_from_clebsch(eq, L, M, N, zeta_0=0.0, clebsch=None, **kwargs): May also be an array of non-uniform coordinates. M : int Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. A good choice is ``m``. If the poloidal stream - function condenses the Fourier spectrum of |B| significantly, then a - larger number may be beneficial. For near omnigenous configurations, - low resolution will be sufficient because (∂‖B‖/∂α)|ρ,ζ vanishes. + Preferably power of 2. A good choice is double ``m``. If the poloidal + stream function condenses the Fourier spectrum of |B| significantly, + then a larger number may be beneficial. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. - zeta_0 : float - ζ coordinate to begin branch cut of (α, ζ) spline. - Default is 0. clebsch : jnp.ndarray Optional, Clebsch coordinate tensor-product grid (ρ, α, ζ). - ``FourierChebyshevBasis.nodes(M,N,L,domain=(zeta_0,zeta_0+2*jnp.pi))``. + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. kwargs Additional parameters to supply to the coordinate mapping function. See ``desc.equilibrium.Equilibrium.map_coordinates``. @@ -403,10 +389,7 @@ def desc_from_clebsch(eq, L, M, N, zeta_0=0.0, clebsch=None, **kwargs): """ if clebsch is None: clebsch = FourierChebyshevBasis.nodes( - check_posint(M), - check_posint(N), - L, - domain=(zeta_0, zeta_0 + 2 * jnp.pi), + check_posint(M), check_posint(N), L, domain=(0, 2 * jnp.pi) ) desc_coords = eq.map_coordinates( coords=clebsch, @@ -614,8 +597,7 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) ) - theta = self._T.eval1d(zeta) - Q = jnp.stack([theta, zeta], axis=-1) + Q = jnp.stack([self._T.eval1d(zeta), zeta], axis=-1) b_sup_z = irfft2_non_uniform( xq=Q, @@ -637,7 +619,6 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): num_rho = shape[-3] shape[0] = num_rho shape[-3] = num_pitch - f.insert(0, theta) _check_interp( # num_alpha, num_rho, num_pitch, num_well, num_quad (1, *shape), diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 7216e352f..202a498ed 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -69,7 +69,8 @@ def cheb_pts(N, lobatto=False, domain=(-1, 1)): def fourier_pts(M): """Get ``M`` Fourier points.""" m = jnp.arange(1, M + 1) - return -jnp.pi + 2 * jnp.pi * m / M + # [0, 2π] instead of [-π, π] to match our choice of α + return 2 * jnp.pi * m / M def harmonic(a, M, axis=-1): diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 2c989cd6f..3695a80e5 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1548,21 +1548,18 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - zeta_0 = 0 # np.pi / data["iota"].item() bounce = Bounce2D( grid=grid, data=grid_data, desc_from_clebsch=Bounce2D.desc_from_clebsch( eq, data["rho"], - M=64, - N=64, - zeta_0=zeta_0, - iota=jnp.broadcast_to(data["iota"], shape=(64**2)), + M=512, + N=32, + iota=jnp.broadcast_to(data["iota"], shape=(32 * 512)), ), - alpha=data["alpha"], - zeta_0=zeta_0, - num_transit=3, + alpha=data["alpha"] - 2 * np.pi * data["iota"], + num_transit=4, Bref=data["Bref"], Lref=data["a"], check=True, From c348378e2f3c577a6c1c61f7773f9941a0e813eb Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 19 Sep 2024 04:45:39 -0400 Subject: [PATCH 09/44] fix test --- tests/test_integrals.py | 61 +++++++++++++++++------------------------ 1 file changed, 25 insertions(+), 36 deletions(-) diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 3695a80e5..bfa0c4472 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1336,13 +1336,13 @@ def drift_analytic(data): @staticmethod def drift_num_integrand(cvdrift, gbdrift, B, pitch): """Integrand of numerator of bounce averaged binormal drift.""" - g = jnp.sqrt(jnp.abs(1 - pitch * B)) + g = jnp.sqrt(1 - pitch * B) return (cvdrift * g) - (0.5 * g * gbdrift) + (0.5 * gbdrift / g) @staticmethod def drift_den_integrand(B, pitch): """Integrand of denominator of bounce averaged binormal drift.""" - return 1 / jnp.sqrt(jnp.abs(1 - pitch * B)) + return 1 / jnp.sqrt(1 - pitch * B) @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) @@ -1473,38 +1473,27 @@ def _cheb_intersect(cheb, k): cheb = cheb.copy() cheb[0] = cheb[0] - k roots = chebroots(cheb) - intersect = roots[ - np.logical_and(np.isreal(roots), np.abs(roots.real) <= 1) - ].real + intersect = roots[np.logical_and(np.isreal(roots), np.abs(roots.real) < 1)].real return intersect - @staticmethod - def _periodic_fun(nodes, M, N): - alpha, zeta = nodes.T - f = -2 * np.cos(1 / (0.1 + zeta**2)) + 2 - return f.reshape(M, N) - @pytest.mark.unit def test_z1_first(self): """Test that bounce points are computed correctly.""" - M, N = 1, 10 - domain = (-1, 1) - nodes = FourierChebyshevBasis.nodes(M, N, domain=domain) - f = -self._periodic_fun(nodes, M, N) - cheb = FourierChebyshevBasis(f, domain=domain).compute_cheb(fourier_pts(M)) - pitch = 1 / np.linspace(1, 4, 1) - z1, z2 = cheb.intersect1d(1 / pitch, num_intersect=1) - print(z1) - print(z2) - cheb.check_intersect1d(z1, z2, 1 / pitch) - z1, z2 = TestBounce1DPoints.filter(z1, z2) def f(z): return -2 * np.cos(1 / (0.1 + z**2)) + 2 - r = self._cheb_intersect(chebinterpolate(f, N), 1 / pitch) - np.testing.assert_allclose(z1, r[::2], rtol=1e-3) - np.testing.assert_allclose(z2, r[1::2], rtol=1e-3) + M, N = 1, 10 + alpha, zeta = FourierChebyshevBasis.nodes(M, N).T + cheb = FourierChebyshevBasis(f(zeta).reshape(M, N)).compute_cheb(fourier_pts(M)) + pitch_inv = 3 + z1, z2 = cheb.intersect1d(pitch_inv) + cheb.check_intersect1d(z1, z2, pitch_inv) + z1, z2 = TestBounce1DPoints.filter(z1, z2) + + r = self._cheb_intersect(chebinterpolate(f, N - 1), pitch_inv) + np.testing.assert_allclose(z1, r[np.isclose(r, -0.24, atol=1e-1)]) + np.testing.assert_allclose(z2, r[np.isclose(r, 0.24, atol=1e-1)]) class TestBounce2D: @@ -1519,13 +1508,10 @@ class TestBounce2D: ], ) def test_alpha_sequence(self, alpha_0, iota, num_period, period): - """Test field line poloidal label tracking.""" - # skeleton test to be done later when one zeta != phi. + """Test field line labels.""" iota = np.atleast_1d(iota) alphas = get_alpha(alpha_0, iota, num_period, period) assert alphas.shape == (iota.size, num_period) - for i in range(iota.size): - assert np.unique(alphas[i]).size == num_period, f"{iota} is irrational" print(alphas) @pytest.mark.unit @@ -1538,9 +1524,7 @@ def test_binormal_drift_bounce2d(self): # Recompute on non-symmetric, fft compatible grid. eq = things["eq"] - grid = LinearGrid( - rho=data["rho"], M=eq.M_grid, N=eq.N_grid, NFP=eq.NFP, sym=False - ) + grid = LinearGrid(rho=data["rho"], M=40, N=40, NFP=eq.NFP, sym=False) grid_data = eq.compute( names=Bounce2D.required_names + ["cvdrift", "gbdrift"], grid=grid ) @@ -1554,11 +1538,15 @@ def test_binormal_drift_bounce2d(self): desc_from_clebsch=Bounce2D.desc_from_clebsch( eq, data["rho"], + # if this doesn't indicate bug, then should just transform + # everything to FourierChebyshev because convergence is + # much quicker (as shown by |B|) than double fourier with + # FourierChebyshev theta argument. M=512, N=32, iota=jnp.broadcast_to(data["iota"], shape=(32 * 512)), ), - alpha=data["alpha"] - 2 * np.pi * data["iota"], + alpha=data["alpha"], # - 2 * np.pi * data["iota"], num_transit=4, Bref=data["Bref"], Lref=data["a"], @@ -1585,9 +1573,10 @@ def test_binormal_drift_bounce2d(self): drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg - # np.testing.assert_allclose( # noqa: E800 - # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 - # ) # noqa: E800 + + np.testing.assert_allclose( # noqa: E800 + drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 + ) # noqa: E800 fig, ax = plt.subplots() ax.plot(pitch_inv, drift_analytic) From 258dad0c8f0104d12f92e32c681170486e32be75 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 19 Sep 2024 05:18:32 -0400 Subject: [PATCH 10/44] ----- --- tests/test_integrals.py | 19 +++++++++++++++---- 1 file changed, 15 insertions(+), 4 deletions(-) diff --git a/tests/test_integrals.py b/tests/test_integrals.py index bfa0c4472..04cb4f319 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1524,7 +1524,18 @@ def test_binormal_drift_bounce2d(self): # Recompute on non-symmetric, fft compatible grid. eq = things["eq"] - grid = LinearGrid(rho=data["rho"], M=40, N=40, NFP=eq.NFP, sym=False) + grid = LinearGrid( + rho=data["rho"], M=eq.M_grid, N=eq.N_grid, sym=False, NFP=eq.NFP + ) + # noqa: E800 + # grid = Grid.create_meshgrid( # noqa: E800 + # [ # noqa: E800 + # data["rho"], # noqa: E800 + # fourier_pts(eq._M_grid), # noqa: E800 + # fourier_pts(eq.N_grid + 1) / eq.NFP, # noqa: E800 + # ], # noqa: E800 + # NFP=eq.NFP, # noqa: E800 + # ) # noqa: E800 grid_data = eq.compute( names=Bounce2D.required_names + ["cvdrift", "gbdrift"], grid=grid ) @@ -1538,9 +1549,9 @@ def test_binormal_drift_bounce2d(self): desc_from_clebsch=Bounce2D.desc_from_clebsch( eq, data["rho"], - # if this doesn't indicate bug, then should just transform - # everything to FourierChebyshev because convergence is - # much quicker (as shown by |B|) than double fourier with + # if this doesn't indicate bug with numpy, then should just + # transform everything to FourierChebyshev because convergence + # is much quicker (as shown by |B|) than double fourier with # FourierChebyshev theta argument. M=512, N=32, From 22df2242f263aa317c9752b876aefd5f5d09a094 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 19 Sep 2024 17:20:11 -0400 Subject: [PATCH 11/44] Document solution to discontinuity --- desc/integrals/bounce_integral.py | 52 ++++++++++++++++--------------- tests/test_integrals.py | 24 +++++++------- 2 files changed, 40 insertions(+), 36 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index b2539414a..287961ede 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -108,6 +108,13 @@ def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): return T, B +def _swap_pl(f): + # Given shape (L, num_pitch, -1) or (num_pitch, L, -1) or (num_pitch, -1) + # swap L and num_pitch axes. + assert f.ndim <= 3 + return jnp.swapaxes(f, 0, -2) + + # TODO: After GitHub issue #1034 is resolved, we should pass in the previous # θ(α) coordinates as an initial guess for the next coordinate mapping. # Perhaps tell the optimizer to perturb the coefficients of the @@ -191,17 +198,22 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) coordinate ϕ, but this balloons the frequency, and hence the degree of the series. - (Although, because Fourier series may converge faster than Chebyshev, - an alternate strategy is to interpolate |B| to a double Fourier series in + One may note that because Fourier series converge faster than Chebyshev, + an alternate strategy is to interpolate |B| to a double Fourier series in (ϑ, ϕ), then apply bisection methods to find roots of f with mesh size inversely proportional to the max frequency along the field line: M ι + N. ``Bounce2D`` does not use that approach because that root-finding scheme is inferior. Note that if the Chebyshev series convergence is not satisfactory, - an option with faster convergence would be to use a filtered Fourier series + we can implement a change of variables for the Chebyshev interpolation that + may allow earlier truncation of the series without loss of accuracy. Another + option with faster convergence still is to use a filtered Fourier series with a non-periodic basis in the Gibbs layer. An example is given at - doi.org/10.1016/j.aml.2006.10.001. Before choosing that option however, - one should test a few change of variables for the Chebyshev interpolation - that may allow earlier truncation of the series without loss of accuracy.) + doi.org/10.1016/j.aml.2006.10.001. Empirically, we observe that the convergence + of the Chebyshev series is satisfactory. It is particularly fast for θ(α, ζ). + Furthermore, for near omnigenous configurations, Fourier series converges rapidly + for |B|(α, ζ) because (∂‖B‖/∂α)|ρ,ζ vanishes. However, the Fourier series of + θ(α, ζ) converges slower than desired; we must accept this because no basis + supports a larger convergence region in the complex plane than do Fourier series. After obtaining the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. @@ -366,9 +378,8 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): May also be an array of non-uniform coordinates. M : int Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. A good choice is double ``m``. If the poloidal - stream function condenses the Fourier spectrum of |B| significantly, - then a larger number may be beneficial. + Preferably power of 2. A good choice is double ``m``. If the Fourier + spectrum of θ is wide, then a larger number may be beneficial. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. @@ -420,13 +431,6 @@ def reshape_data(grid, *arys): f = [grid.meshgrid_reshape(d, "rtz") for d in arys] return f if len(f) > 1 else f[0] - @staticmethod - def _swap_pl(f): - # Given shape (L, num_pitch, -1) or (num_pitch, L, -1) or (num_pitch, -1) - # swap L and num_pitch axes. - assert f.ndim <= 3 - return jnp.swapaxes(f, 0, -2) - def points(self, pitch_inv, *, num_well=None): """Compute bounce points. @@ -464,9 +468,7 @@ def points(self, pitch_inv, *, num_well=None): pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) - z1 = Bounce2D._swap_pl(z1) - z2 = Bounce2D._swap_pl(z2) - return z1, z2 + return _swap_pl(z1), _swap_pl(z2) def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """Check that bounce points are computed correctly. @@ -496,8 +498,8 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """ return self._B.check_intersect1d( - z1=Bounce2D._swap_pl(z1), - z2=Bounce2D._swap_pl(z2), + z1=_swap_pl(z1), + z2=_swap_pl(z2), k=atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T, plot=plot, **_set_default_plot_kwargs(kwargs), @@ -576,7 +578,6 @@ def integrate( # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) - result = Bounce2D._swap_pl(result) return result def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): @@ -608,7 +609,7 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): ) B = self._B.eval1d(zeta) f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] - result = ( + result = _swap_pl( (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) .reshape(shape) .dot(self._w) @@ -622,8 +623,9 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): _check_interp( # num_alpha, num_rho, num_pitch, num_well, num_quad (1, *shape), - *map(Bounce2D._swap_pl, (zeta, b_sup_z, B, result)), - list(map(Bounce2D._swap_pl, f)), + *map(_swap_pl, (zeta, b_sup_z, B)), + result, + list(map(_swap_pl, f)), plot, ) return result diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 04cb4f319..fc5dc0c9e 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1543,19 +1543,21 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. + M, N = 512, 8 + # Neo does 200 x 200 for their algorithm, so I suppose this isn't too weird. + # I have concern that numpy computes Nyquist frequency component incorrectly, + # and the reason high Fourier resolution is required is that with large sample + # frequency the incorrect max frequency component is higher than the max + # frequency of theta, hiding the mistake. bounce = Bounce2D( grid=grid, data=grid_data, desc_from_clebsch=Bounce2D.desc_from_clebsch( eq, data["rho"], - # if this doesn't indicate bug with numpy, then should just - # transform everything to FourierChebyshev because convergence - # is much quicker (as shown by |B|) than double fourier with - # FourierChebyshev theta argument. - M=512, - N=32, - iota=jnp.broadcast_to(data["iota"], shape=(32 * 512)), + M=M, + N=N, + iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), alpha=data["alpha"], # - 2 * np.pi * data["iota"], num_transit=4, @@ -1584,10 +1586,10 @@ def test_binormal_drift_bounce2d(self): drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg - - np.testing.assert_allclose( # noqa: E800 - drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 - ) # noqa: E800 + # + # np.testing.assert_allclose( # noqa: E800 + # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 + # ) # noqa: E800 fig, ax = plt.subplots() ax.plot(pitch_inv, drift_analytic) From 278168135252a3f5fb3564a34765efdf53adae47 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 19 Sep 2024 17:51:21 -0400 Subject: [PATCH 12/44] Fix comment --- desc/integrals/bounce_integral.py | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 287961ede..e09060726 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -208,12 +208,13 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) may allow earlier truncation of the series without loss of accuracy. Another option with faster convergence still is to use a filtered Fourier series with a non-periodic basis in the Gibbs layer. An example is given at - doi.org/10.1016/j.aml.2006.10.001. Empirically, we observe that the convergence - of the Chebyshev series is satisfactory. It is particularly fast for θ(α, ζ). - Furthermore, for near omnigenous configurations, Fourier series converges rapidly - for |B|(α, ζ) because (∂‖B‖/∂α)|ρ,ζ vanishes. However, the Fourier series of - θ(α, ζ) converges slower than desired; we must accept this because no basis - supports a larger convergence region in the complex plane than do Fourier series. + doi.org/10.1016/j.aml.2006.10.001. Empirically, we observe that the + Chebyshev series has satisfactory convergence. It is particularly fast for + θ(α, ζ). Furthermore, the Fourier series converges rapidly for |B|(α, ζ) + in near omnigenous configurations because (∂|B|/∂α)|ρ,ζ vanishes. However, + the Fourier series of θ(α, ζ) converges slower than desired; we must accept + this because no basis supports a larger convergence region in the complex + plane than do Fourier series. After obtaining the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. @@ -247,7 +248,7 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) """ required_names = ["B^zeta", "|B|", "iota"] - get_pitch_inv = staticmethod(get_pitch_inv) + get_pitch_inv_quad = staticmethod(get_pitch_inv) def __init__( self, @@ -378,8 +379,8 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): May also be an array of non-uniform coordinates. M : int Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. A good choice is double ``m``. If the Fourier - spectrum of θ is wide, then a larger number may be beneficial. + Preferably power of 2. If the Fourier spectrum of θ is wide, then + a large number is recommended. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. From 41d440ab3fff26d7da7e316c914195b5d21a371b Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 21 Sep 2024 05:42:31 -0400 Subject: [PATCH 13/44] Remove unnecessary interpolation: MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit For unoptimized |B|, typically higher N is needed to capture all oscillations. θ will likely always be fine with low N since it's essentially just proportional to iota at fixed alpha, so Chebyshev converges fast. Rather than doing coordinate mapping on M * N tensor grid each large, do it on M * (N_{\theta} where N θ is small. Then use the resulting, accurate, θ(α, ζ) to compute the interpolation points in (α, ζ) so that |B| (α, ζ) can be constructed with large Chebyshev resolution. This reduces the number of points to perform root finding on each time we want to compute a neoclassical objective, decreases interpolation error, increases speed etc. --- desc/integrals/basis.py | 28 ++- desc/integrals/bounce_integral.py | 317 +++++++++++++++++------------- desc/integrals/interp_utils.py | 26 ++- tests/test_integrals.py | 11 +- tests/test_interp_utils.py | 8 +- 5 files changed, 240 insertions(+), 150 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 2ed91e5ec..956e3ddcc 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -13,6 +13,7 @@ cheb_from_dct, cheb_pts, chebroots_vec, + dct_from_cheb, fourier_pts, harmonic, idct_non_uniform, @@ -185,7 +186,7 @@ def nodes(M, N, L=None, domain=(-1, 1), lobatto=False): """ x = fourier_pts(M) - y = cheb_pts(N, lobatto, domain) + y = cheb_pts(N, domain, lobatto) if L is None: coords = (x, y) else: @@ -234,7 +235,7 @@ def harmonics(self): Real valued spectral coefficients for Fourier-Chebyshev basis. """ - a_mn = harmonic(cheb_from_dct(self._c, axis=-1), self.M, axis=-2) + a_mn = harmonic(cheb_from_dct(self._c), self.M, axis=-2) assert a_mn.shape[-2:] == (self.M, self.N) return a_mn @@ -254,7 +255,7 @@ def compute_cheb(self, x): """ # Add axis to broadcast against Chebyshev coefficients. x = jnp.atleast_1d(x)[..., jnp.newaxis] - cheb = cheb_from_dct(irfft_non_uniform(x, self._c, self.M, axis=-2), axis=-1) + cheb = cheb_from_dct(irfft_non_uniform(x, self._c, self.M, axis=-2)) assert cheb.shape[-2:] == (x.shape[-2], self.N) return ChebyshevBasisSet(cheb, self.domain) @@ -305,6 +306,27 @@ def N(self): """Chebyshev spectral resolution.""" return self.cheb.shape[-1] + def evaluate(self, N): + """Evaluate Chebyshev series at N Chebyshev points. + + Evaluate each function in this set + { fₓ | fₓ : y ↦ ∑ₙ₌₀ᴺ⁻¹ aₙ(x) Tₙ(y) } + at y points given by the N Chebyshev points. + + Parameters + ---------- + N : int + Grid resolution in y direction. Preferably power of 2. + + Returns + ------- + fq : jnp.ndarray + Shape (..., M, N) + Chebyshev series evaluated at N Chebyshev points. + + """ + return idct(dct_from_cheb(self.cheb), type=2, n=N, axis=-1) * N + def isomorphism_to_C1(self, y): """Return coordinates z ∈ ℂ isomorphic to (x, y) ∈ ℂ². diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index e09060726..e4f2ca472 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -5,7 +5,7 @@ from numpy.fft import rfft2 from orthax.legendre import leggauss -from desc.backend import jnp +from desc.backend import dct, jnp from desc.integrals.basis import ChebyshevBasisSet, FourierChebyshevBasis from desc.integrals.bounce_utils import ( _bounce_quadrature, @@ -18,7 +18,13 @@ interp_to_argmin, plot_ppoly, ) -from desc.integrals.interp_utils import interp_rfft2, irfft2_non_uniform, polyder_vec +from desc.integrals.interp_utils import ( + cheb_from_dct, + cheb_pts, + interp_rfft2, + irfft2_non_uniform, + polyder_vec, +) from desc.integrals.quad_utils import ( automorphism_sin, bijection_from_disc, @@ -44,6 +50,7 @@ def _transform_to_desc(grid, f, is_reshaped=False): grid : Grid Tensor-product grid in (θ, ζ) with uniformly spaced nodes in (2π × 2π) poloidal and toroidal coordinates. + Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. f : jnp.ndarray Function evaluated on ``grid``. @@ -57,54 +64,95 @@ def _transform_to_desc(grid, f, is_reshaped=False): if not is_reshaped: f = grid.meshgrid_reshape(f, "rtz") # real fft over poloidal since usually m > n - a = rfft2(f, axes=(-1, -2), norm="forward") - assert a.shape[-2:] == (grid.num_theta // 2 + 1, grid.num_zeta) - return a + return rfft2(f, axes=(-1, -2), norm="forward") -def _transform_to_clebsch(grid, desc_from_clebsch, B, is_reshaped=False): +def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): """Transform to Clebsch spectral domain. Parameters ---------- grid : Grid - Tensor-product grid in (ρ, θ, ζ) with uniformly spaced nodes in + Tensor-product grid in (θ, ζ) with uniformly spaced nodes in + (2π × 2π) poloidal and toroidal coordinates. + Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. + nodes : jnp.ndarray + Shape (L, M, N, 2) or (M, N, 2). + DESC coordinates (θ, ζ) sourced from the Clebsch coordinates + ``FourierChebyshevBasis.nodes(M,N,domain=(0,2*jnp.pi))``. + f : jnp.ndarray + Function evaluated on ``grid``. + + Returns + ------- + a : FourierChebyshevBasis + Spectral coefficients of f(α, ζ). + + """ + assert nodes.shape[-1] == 2 + if not is_reshaped: + f = grid.meshgrid_reshape(f, "rtz") + + M, N = nodes.shape[-3], nodes.shape[-2] + return FourierChebyshevBasis( + f=interp_rfft2( + # Interpolate to nodes in Clebsch space, + # which is not a tensor product node set in DESC space. + xq=nodes.reshape(*nodes.shape[:-3], M * N, 2), + f=f[..., jnp.newaxis, :, :], + axes=(-1, -2), + ).reshape(*nodes.shape[:-3], M, N), + domain=(0, 2 * jnp.pi), + ) + + +def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): + """Transform to single variable Clebsch spectral domain. + + Parameters + ---------- + grid : Grid + Tensor-product grid in (θ, ζ) with uniformly spaced nodes in (2π × 2π) poloidal and toroidal coordinates. - Note that below shape notation defines - L = ``grid.num_rho``, m = ``grid.num_theta``, and n = ``grid.num_zeta``. - Preferably power of 2 for ``m`` and ``n``. - desc_from_clebsch : jnp.ndarray - Shape (L, M, N, 3). - DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. + alpha : jnp.ndarray + Shape (L, num_transit) or (num_transit, ). + Sequence of poloidal coordinates A = (α₀, α₁, …, αₘ₋₁) that specify field line. + theta : jnp.ndarray + Shape (L, M, N) or (M, N). + DESC coordinates θ sourced from the Clebsch coordinates + ``FourierChebyshevBasis.nodes(M,N,domain=(0,2*jnp.pi))``. B : jnp.ndarray |B| evaluated on ``grid``. + N_B : int + Desired Chebyshev spectral resolution for |B|. Preferably power of 2. Returns ------- - T, B : (FourierChebyshevBasis, FourierChebyshevBasis) - Spectral coefficients of θ(α, ζ) and |B|(α, ζ). + T, B : (ChebyshevBasisSet, ChebyshevBasisSet) + Set of 1D Chebyshev spectral coefficients of |B| along field line. + {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the + sequence of poloidal coordinates that specify the field line. + Likewise with θ. """ - domain = (0, 2 * jnp.pi) - M, N = desc_from_clebsch.shape[-3:-1] - # θ is computed on the optimal nodes in Clebsch space, - # which is a tensor product node set in Clebsch space. - T = FourierChebyshevBasis(f=desc_from_clebsch[..., 1], domain=domain) - if not is_reshaped: B = grid.meshgrid_reshape(B, "rtz") - # Interpolate to optimal nodes in Clebsch space, - # which is not a tensor product node set in DESC space. + + # Evaluating set of single variable maps is more efficient than evaluating + # multivariable map, so we project θ to a set of Chebyshev series. + T = FourierChebyshevBasis(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) + theta = T.evaluate(N_B) + xq = jnp.stack( + [theta, jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape)], axis=-1 + ).reshape(*alpha.shape[:-1], alpha.shape[-1] * N_B, 2) B = interp_rfft2( - xq=desc_from_clebsch[..., 1:].reshape(-1, M * N, 2), + xq=xq, f=B[..., jnp.newaxis, :, :], axes=(-1, -2), - ).reshape(-1, M, N) - B = FourierChebyshevBasis( - f=B if desc_from_clebsch.ndim == 4 else B.squeeze(axis=0), - domain=domain, - ) + ).reshape(*alpha.shape, N_B) + # Need |B| parameterized by single variable to compute roots. + B = ChebyshevBasisSet(cheb_from_dct(dct(B, type=2, axis=-1)) / N_B, T.domain) return T, B @@ -116,10 +164,9 @@ def _swap_pl(f): # TODO: After GitHub issue #1034 is resolved, we should pass in the previous -# θ(α) coordinates as an initial guess for the next coordinate mapping. +# θ(α, ζ) coordinates as an initial guess for the next coordinate mapping. # Perhaps tell the optimizer to perturb the coefficients of the -# |B|(α, ζ) directly? think perturbing alpha is equivalent to perturbing -# lambda. Not sure if possible.. +# θ(α, ζ) directly? think this is equivalent to perturbing lambda. class Bounce2D(IOAble): @@ -140,56 +187,68 @@ class Bounce2D(IOAble): Notes ----- - Brief motivation and description of algorithm for developers. + Brief description of algorithm. + + Magnetic field line with label α, defined by B = ∇ρ × ∇α, is determined from + α : ρ, θ, ζ ↦ θ + λ(ρ,θ,ζ) − ι(ρ) [ζ + ω(ρ,θ,ζ)] + Interpolate Fourier-Chebyshev series to DESC poloidal coordinate. + θ : α, ζ ↦ tₘₙ exp(jmα) Tₙ(ζ) + |B| : α, ζ ↦ bₙ(θ(α, ζ)) Tₙ(ζ) + Compute bounce points. + r(ζₖ) = |B|(ζₖ) − 1/λ = 0 + Interpolate smooth components of integrand with FFTs. + G : α, ζ ↦ gₘₙ exp(j [m θ(α,ζ) + n ζ] ) + Perform Gaussian quadrature after removing singularities. + Fᵢ : λ, ζ₁, ζ₂ ↦ ∫ᵢ f(λ, ζ, {Gⱼ}) dζ + + Longer description for developers. For applications which reduce to computing a nonlinear function of distance along field lines between bounce points, it is required to identify these points with field-line-following coordinates. (In the special case of a linear function summing integrals between bounce points over a flux surface, arbitrary - coordinate systems may be used as this operation reduces to a surface integral, + coordinate systems may be used as that task reduces to a surface integral, which is invariant to the order of summation). The DESC coordinate system is related to field-line-following coordinate systems by a relation whose solution is best found with Newton iteration. - There is a unique real solution to this equation, so Newton iteration is a + There is a unique real solution to that relation, so Newton iteration is a globally convergent root-finding algorithm here. For the task of finding - bounce points, even if the inverse map: θ(α, ζ) was known, Newton iteration - is not a globally convergent algorithm to find the real roots of - f : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following coordinate. - For this, function approximation of |B| is necessary. + bounce points, Newton iteration is not a globally convergent algorithm to + find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following + coordinate. For this, function approximation of |B| is necessary. Therefore, to compute bounce points {(ζ₁, ζ₂)}, we approximate |B| by a - series expansion of basis functions in (α, ζ) coordinates, restricting the - class of basis functions to low order (e.g. N = 2ᵏ where k is small) - algebraic or trigonometric polynomial with integer frequencies. These are - the two classes useful for function approximation and for which there exists - globally convergent root-finding algorithms. We require low order because - the computation expenses grow with the number of potential roots, and the - theorem of algebra states that number is N (2N) for algebraic + series expansion of basis functions parameterized by a single variable ζ, + restricting the class of basis functions to low order (e.g. N = 2ᵏ where + k is small) algebraic or trigonometric polynomial with integer frequencies. + These are the two classes useful for function approximation and for which + there exists globally convergent root-finding algorithms. We require low + order because the computation expenses grow with the number of potential + roots, and the theorem of algebra states that number is N (2N) for algebraic (trigonometric) polynomials of degree N. The frequency transform of a map under the chosen basis must be concentrated - at low frequencies for the series to converge to the true function fast. - For periodic (non-periodic) maps, the best basis is a Fourier (Chebyshev) - series. Both converge exponentially, but the larger region of convergence in - the complex plane of Fourier series make it preferable in practice to choose - coordinate systems such that the function to approximate is periodic. The - Chebyshev series is preferred to other orthogonal polynomial series since + at low frequencies for the series to converge fast. For periodic + (non-periodic) maps, the best basis is a Fourier (Chebyshev) series. Both + converge exponentially, but the larger region of convergence in the complex + plane of Fourier series make it preferable in practice to choose coordinate + systems such that the function to approximate is periodic. The Chebyshev + polynomials are preferred to other orthogonal polynomial series since fast discrete polynomial transforms (DPT) are implemented via fast transform to Chebyshev then DCT. Although nothing prohibits a direct DPT, we want to - rely on existing, optimized libraries. There are other reasons to prefer - Chebyshev series not discussed here. Therefore, |B| is interpolated to a - Fourier-Chebyshev series in (α, ζ). + rely on existing libraries. There are other reasons to prefer Chebyshev series + not discussed here. Therefore, a Fourier-Chebyshev series is chosen to + interpolate θ(α,ζ), and a piecewise Chebyshev series interpolates |B|(ζ). Computing accurate series expansions in (α, ζ) coordinates demands particular interpolation points in that coordinate system. Newton iteration - is used to compute θ at these interpolation points. Note that interpolation - is necessary because there is no transformation that converts series - coefficients in periodic coordinates, e.g. (ϑ, ϕ), to a low order - polynomial basis in non-periodic coordinates. For example, one can obtain - series coefficients in (α, ϕ) coordinates from those in (ϑ, ϕ) as follows + is used to compute θ at these points. Note that interpolation is necessary + because there is no transformation that converts series coefficients in + periodic coordinates, e.g. (ϑ, ϕ), to a low order polynomial basis in + non-periodic coordinates. For example, one can obtain series coefficients in + (α, ϕ) coordinates from those in (ϑ, ϕ) as follows g : ϑ, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mϑ + nϕ]) - g : α, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mα + (m ι + n)ϕ]) However, the basis for the latter are trigonometric functions with irrational frequencies, courtesy of the irrational rotational transform. @@ -198,35 +257,29 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) coordinate ϕ, but this balloons the frequency, and hence the degree of the series. - One may note that because Fourier series converge faster than Chebyshev, - an alternate strategy is to interpolate |B| to a double Fourier series in - (ϑ, ϕ), then apply bisection methods to find roots of f with mesh size - inversely proportional to the max frequency along the field line: M ι + N. - ``Bounce2D`` does not use that approach because that root-finding scheme is - inferior. Note that if the Chebyshev series convergence is not satisfactory, - we can implement a change of variables for the Chebyshev interpolation that - may allow earlier truncation of the series without loss of accuracy. Another - option with faster convergence still is to use a filtered Fourier series - with a non-periodic basis in the Gibbs layer. An example is given at - doi.org/10.1016/j.aml.2006.10.001. Empirically, we observe that the - Chebyshev series has satisfactory convergence. It is particularly fast for - θ(α, ζ). Furthermore, the Fourier series converges rapidly for |B|(α, ζ) - in near omnigenous configurations because (∂|B|/∂α)|ρ,ζ vanishes. However, - the Fourier series of θ(α, ζ) converges slower than desired; we must accept - this because no basis supports a larger convergence region in the complex - plane than do Fourier series. - - After obtaining the bounce points, the supplied quadrature is performed. + Recall that periodicity enables faster convergence, motivating the desire + to instead interpolate |B|(ϑ, ϕ) with a double Fourier series and applying + bisection methods to find bounce points with mesh size inversely + proportional to the max frequency along the field line: M ι + N. ``Bounce2D`` + does not use that approach as that root-finding scheme is inferior. + The reason θ is not interpolated with a double Fourier series θ(ϑ, ζ) is + because quadrature points along |B|(α=α₀, ζ) can be identified by a single + variable; evaluating the multivariable map θ(ϑ(α, ζ), ζ) is expensive + compared to evaluating the single variable map θ(α=α₀, ζ). + Another option is to use a filtered Fourier series, + doi.org/10.1016/j.aml.2006.10.001. Empirically the Chebyshev series + θ(α=α₀, ζ) converges fast. + + After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. Fast fourier transforms interpolate functions in the integrand to the quadrature nodes. - Fast transforms are used where possible, though fast multipoint methods - are not yet implemented. For non-uniform interpolation, Vandermode MMT with - the linear algebra libraries of JAX are used. It should be worthwhile to use - the inverse non-uniform fast transforms. Fast multipoint methods are - preferable because they are exact, but this requires more development work. - Future work may implement these techniques. + Fast transforms are used where possible. Fast multipoint methods are not + implemented. For non-uniform interpolation, MMTs are used. It should be + worthwhile to use the inverse non-uniform fast transforms, so long as the + quadrature packs nodes at reasonable density. Fast multipoint methods are + preferable because they are exact, but that requires more development work. See Also -------- @@ -236,10 +289,6 @@ class of basis functions to low order (e.g. N = 2ᵏ where k is small) which the root-finding must be done to map from DESC to Clebsch coords is fixed to ``L*M*N``, independent of the number of toroidal transits. - Warnings - -------- - It is currently assumed that ζ = ϕ. - Attributes ---------- required_names : list @@ -254,11 +303,12 @@ def __init__( self, grid, data, - desc_from_clebsch, + theta, # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb alpha=0.0, num_transit=32, + N_B=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), Bref=1.0, @@ -285,14 +335,16 @@ def __init__( data : dict[str, jnp.ndarray] Data evaluated on ``grid``. Must include names in ``Bounce2D.required_names``. - desc_from_clebsch : jnp.ndarray - Shape (L, M, N, 3). - DESC coordinates (ρ, θ, ζ) sourced from the Clebsch coordinates + theta : jnp.ndarray + Shape (L, M, N). + DESC coordinates θ sourced from the Clebsch coordinates ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. alpha : float Starting field line poloidal label. num_transit : int Number of toroidal transits to follow field line. + N_B : int + Desired Chebyshev spectral resolution for |B|. Preferably power of 2. quad : (jnp.ndarray, jnp.ndarray) Quadrature points xₖ and weights wₖ for the approximate evaluation of an integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). Default is 32 points. @@ -335,16 +387,9 @@ def __init__( self._n = grid.num_zeta self._x, self._w = get_quadrature(quad, automorphism) - # Compute spectral coefficients. - self._b_sup_z = _transform_to_desc( - grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped - ) - T, B = _transform_to_clebsch( - grid, desc_from_clebsch, data["|B|"] / Bref, is_reshaped - ) # peel off field lines iota = data["iota"].ravel() - alphas = get_alpha( + alpha = get_alpha( alpha, iota=( grid.compress(iota) @@ -355,20 +400,28 @@ def __init__( num_transit=num_transit, period=2 * jnp.pi, ) - - # Set of 1D Chebyshev spectral coefficients of |B| along field line. - # {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the - # sequence of poloidal coordinates that specify the field line. - # Evaluating set of Chebyshev series more efficient than evaluating - # Fourier Chebyshev series, so we project θ to Chebyshev series as well. - self._B = B.compute_cheb(alphas) - self._T = T.compute_cheb(alphas) - assert self._B.M == self._T.M == num_transit - assert self._B.N == self._T.N == desc_from_clebsch.shape[-2] - + # Compute spectral coefficients. + self._T, self._B = _transform_to_clebsch_1d( + grid, alpha, theta, data["|B|"] / Bref, N_B, is_reshaped + ) + self._b_sup_z = _transform_to_desc( + grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped + ) + assert self._T.M == self._B.M == num_transit + assert self._T.N == theta.shape[-1] + assert self._B.N == N_B + + # The Fourier series converges fast for |B|(α, ζ); + # for near omnigenous configurations in particular, (∂|B|/∂α)|ρ,ζ vanishes. + # However, the Fourier series of θ(α, ζ) converges slower than desired. + # Small discontinuities of quantities evaluated between adjacent cuts of a + # field line vanish at M > 256. @staticmethod - def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): - """Return DESC coordinates of optimal Fourier Chebyshev basis nodes. + def compute_theta(eq, L, M=512, N=16, clebsch=None, **kwargs): + """Return DESC coordinates θ of Fourier Chebyshev basis nodes. + + The Fourier spectrum of θ in α is wider than the Chebyshev spectrum + of θ in ζ, so M > N is recommended. Parameters ---------- @@ -379,8 +432,7 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): May also be an array of non-uniform coordinates. M : int Grid resolution in poloidal direction for Clebsch coordinate grid. - Preferably power of 2. If the Fourier spectrum of θ is wide, then - a large number is recommended. + Preferably power of 2. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. @@ -393,23 +445,22 @@ def desc_from_clebsch(eq, L, M, N, clebsch=None, **kwargs): Returns ------- - desc_coords : jnp.ndarray - Shape (L, M, N, 3). - DESC coordinate grid (ρ, θ, ζ) sourced from the Clebsch coordinate - tensor-product grid (ρ, α, ζ). + theta : jnp.ndarray + Shape (L, M, N). + DESC coordinates θ sourced from the Clebsch coordinates + ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. """ if clebsch is None: clebsch = FourierChebyshevBasis.nodes( check_posint(M), check_posint(N), L, domain=(0, 2 * jnp.pi) ) - desc_coords = eq.map_coordinates( + return eq.map_coordinates( coords=clebsch, inbasis=("rho", "alpha", "zeta"), period=(jnp.inf, jnp.inf, jnp.inf), **kwargs, - ).reshape(-1, M, N, 3) - return desc_coords + ).reshape(-1, M, N, 3)[..., 1] @staticmethod def reshape_data(grid, *arys): @@ -693,17 +744,16 @@ class Bounce1D(IOAble): along field lines between bounce points, it is required to identify these points with field-line-following coordinates. (In the special case of a linear function summing integrals between bounce points over a flux surface, arbitrary - coordinate systems may be used as this operation reduces to a surface integral, + coordinate systems may be used as that task reduces to a surface integral, which is invariant to the order of summation). The DESC coordinate system is related to field-line-following coordinate systems by a relation whose solution is best found with Newton iteration. - There is a unique real solution to this equation, so Newton iteration is a + There is a unique real solution to that relation, so Newton iteration is a globally convergent root-finding algorithm here. For the task of finding - bounce points, even if the inverse map: θ(α, ζ) was known, Newton iteration - is not a globally convergent algorithm to find the real roots of - f : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following coordinate. - For this, function approximation of |B| is necessary. + bounce points, Newton iteration is not a globally convergent algorithm to + find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following + coordinate. For this, function approximation of |B| is necessary. The function approximation in ``Bounce1D`` is ignorant that the objects to approximate are defined on a bounded subset of ℝ². Instead, the domain is @@ -711,9 +761,8 @@ class Bounce1D(IOAble): cannot support reconstruction of the function near the origin. As the functions of interest do not vanish at infinity, pseudo-spectral techniques are not used. Instead, function approximation is done with local splines. - This is useful if one can efficiently obtain data along field lines and - most efficient if the number of toroidal transits to follow a field line is - not too large. + This is useful if one can efficiently obtain data along field lines the + number of toroidal transits to follow a field line is not large. After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 202a498ed..2d1c44445 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -29,7 +29,7 @@ # series truncation, not the infinite limit). -def cheb_pts(N, lobatto=False, domain=(-1, 1)): +def cheb_pts(N, domain=(-1, 1), lobatto=False): """Get ``N`` Chebyshev points mapped to given domain. Warnings @@ -45,11 +45,11 @@ def cheb_pts(N, lobatto=False, domain=(-1, 1)): ---------- N : int Number of points. + domain : (float, float) + Domain for points. lobatto : bool Whether to return the Gauss-Lobatto (extrema-plus-endpoint) instead of the interior roots for Chebyshev points. - domain : (float, float) - Domain for points. Returns ------- @@ -323,6 +323,26 @@ def cheb_from_dct(a, axis=-1): return cheb +def dct_from_cheb(cheb, axis=-1): + """Get discrete cosine transform from discrete Chebyshev transform. + + Parameters + ---------- + cheb : jnp.ndarray + Discrete Chebyshev transform coefficients, e.g.``cheb_from_dct(a)``. + axis : int + Axis along which to transform. + + Returns + ------- + a : jnp.ndarray + Chebyshev coefficients along ``axis``. + + """ + a = cheb.copy().at[Index.get(0, axis, cheb.ndim)].multiply(2.0) + return a + + def interp_dct(xq, f, lobatto=False, axis=-1): """Interpolate ``f`` to ``xq`` with discrete Chebyshev transform. diff --git a/tests/test_integrals.py b/tests/test_integrals.py index fc5dc0c9e..fc8e4f274 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -956,7 +956,7 @@ def test_bounce_quadrature(self, is_strong, quad, automorphism): check=True, **kwargs, ) - result = bounce.integrate(integrand, pitch_inv, check=True) + result = bounce.integrate(integrand, pitch_inv, check=True, plot=True) assert np.count_nonzero(result) == 1 np.testing.assert_allclose(result.sum(), truth, rtol=1e-4) @@ -1062,11 +1062,11 @@ class TestBounce1D: @staticmethod def _example_numerator(g_zz, B, pitch): f = (1 - 0.5 * pitch * B) * g_zz - return safediv(f, jnp.sqrt(1 - pitch * B)) + return safediv(f, jnp.sqrt(jnp.abs(1 - pitch * B))) @staticmethod def _example_denominator(B, pitch): - return safediv(1, jnp.sqrt(1 - pitch * B)) + return safediv(1, jnp.sqrt(jnp.abs(1 - pitch * B))) @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d * 4) @@ -1369,7 +1369,6 @@ def test_binormal_drift_bounce1d(self): f=f, num_well=1, check=True, - plot=True, ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, @@ -1543,7 +1542,7 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 512, 8 + M, N = 512, 16 # Neo does 200 x 200 for their algorithm, so I suppose this isn't too weird. # I have concern that numpy computes Nyquist frequency component incorrectly, # and the reason high Fourier resolution is required is that with large sample @@ -1552,7 +1551,7 @@ def test_binormal_drift_bounce2d(self): bounce = Bounce2D( grid=grid, data=grid_data, - desc_from_clebsch=Bounce2D.desc_from_clebsch( + theta=Bounce2D.compute_theta( eq, data["rho"], M=M, diff --git a/tests/test_interp_utils.py b/tests/test_interp_utils.py index 47b4e6226..fddc9f5d0 100644 --- a/tests/test_interp_utils.py +++ b/tests/test_interp_utils.py @@ -176,7 +176,7 @@ def test_cheb_pts(self, N): """Test we use Chebyshev points compatible with standard definition of DCT.""" np.testing.assert_allclose(cheb_pts(N), chebpts1(N)[::-1], atol=1e-15) np.testing.assert_allclose( - cheb_pts(N, lobatto=True, domain=(-np.pi, np.pi)), + cheb_pts(N, domain=(-np.pi, np.pi), lobatto=True), np.pi * chebpts2(N)[::-1], atol=1e-15, ) @@ -266,10 +266,10 @@ def test_dct(self, f, M, lobatto): # Need to test fft used in Fourier Chebyshev interpolation due to issues like # https://github.com/scipy/scipy/issues/15033 # https://github.com/scipy/scipy/issues/21198 - # https://github.com/google/jax/issues/22466, + # https://github.com/google/jax/issues/22466. domain = (0, 2 * np.pi) - m = cheb_pts(M, lobatto, domain) - n = cheb_pts(m.size * 10, lobatto, domain) + m = cheb_pts(M, domain, lobatto) + n = cheb_pts(m.size * 10, domain, lobatto) norm = (n.size - lobatto) / (m.size - lobatto) dct_type = 2 - lobatto From a5c87a43437002c1173c37e01df5a6c8a23e1058 Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 21 Sep 2024 07:09:00 -0400 Subject: [PATCH 14/44] Doing stuff marked todo --- desc/integrals/interp_utils.py | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 2d1c44445..d74fcfcf6 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -24,9 +24,8 @@ # TODO: Transformation to make nodes more uniform Boyd eq. 16.46 pg. 336. -# Have a hunch it won't change locations of complex poles much, so using -# more uniformly spaced nodes could speed up convergence (wrt early -# series truncation, not the infinite limit). +# Have a hunch more uniformly spaced nodes could speed up convergence +# Edit: Seems unnecessary for now. Chebyshev part converges fine. def cheb_pts(N, domain=(-1, 1), lobatto=False): @@ -208,7 +207,6 @@ def irfft_non_uniform(xq, a, n, axis=-1): m = jnp.fft.rfftfreq(n, d=1 / n) basis = jnp.exp(-1j * m * xq[..., jnp.newaxis]) fq = 2.0 * jnp.linalg.vecdot(basis, a).real - # TODO: Test JAX does this optimization automatically. # ℜ〈 basis, a 〉= cos(m xq)⋅ℜ(a) − sin(m xq)⋅ℑ(a) return fq From 4eb6db2aa2a2f12cf69cff955cd61cf511c14e8d Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 21 Sep 2024 07:59:01 -0400 Subject: [PATCH 15/44] testing --- desc/integrals/bounce_integral.py | 11 ++++++----- tests/test_integrals.py | 25 +++++++++++++------------ 2 files changed, 19 insertions(+), 17 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index e4f2ca472..7d91a0960 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -30,6 +30,7 @@ bijection_from_disc, get_quadrature, grad_automorphism_sin, + grad_bijection_from_disc, ) from desc.io import IOAble from desc.utils import ( @@ -267,8 +268,7 @@ class Bounce2D(IOAble): variable; evaluating the multivariable map θ(ϑ(α, ζ), ζ) is expensive compared to evaluating the single variable map θ(α=α₀, ζ). Another option is to use a filtered Fourier series, - doi.org/10.1016/j.aml.2006.10.001. Empirically the Chebyshev series - θ(α=α₀, ζ) converges fast. + doi.org/10.1016/j.aml.2006.10.001. After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. @@ -304,11 +304,11 @@ def __init__( grid, data, theta, + N_B, # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb alpha=0.0, num_transit=32, - N_B=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), Bref=1.0, @@ -339,12 +339,12 @@ def __init__( Shape (L, M, N). DESC coordinates θ sourced from the Clebsch coordinates ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + N_B : int + Desired Chebyshev spectral resolution for |B|. Preferably power of 2. alpha : float Starting field line poloidal label. num_transit : int Number of toroidal transits to follow field line. - N_B : int - Desired Chebyshev spectral resolution for |B|. Preferably power of 2. quad : (jnp.ndarray, jnp.ndarray) Quadrature points xₖ and weights wₖ for the approximate evaluation of an integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). Default is 32 points. @@ -665,6 +665,7 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) .reshape(shape) .dot(self._w) + * grad_bijection_from_disc(z1, z2) ) if check: diff --git a/tests/test_integrals.py b/tests/test_integrals.py index fc8e4f274..9cfeeda29 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1369,6 +1369,7 @@ def test_binormal_drift_bounce1d(self): f=f, num_well=1, check=True, + # plot=True, # noqa: E800 ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, @@ -1523,18 +1524,16 @@ def test_binormal_drift_bounce2d(self): # Recompute on non-symmetric, fft compatible grid. eq = things["eq"] - grid = LinearGrid( - rho=data["rho"], M=eq.M_grid, N=eq.N_grid, sym=False, NFP=eq.NFP - ) - # noqa: E800 - # grid = Grid.create_meshgrid( # noqa: E800 - # [ # noqa: E800 - # data["rho"], # noqa: E800 - # fourier_pts(eq._M_grid), # noqa: E800 - # fourier_pts(eq.N_grid + 1) / eq.NFP, # noqa: E800 - # ], # noqa: E800 - # NFP=eq.NFP, # noqa: E800 - # ) # noqa: E800 + # FIXME: Change LinearGrid to default to Fourier points nodes. Interpolation + # fails on LinearGrid. + grid = Grid.create_meshgrid( + [ + data["rho"], + fourier_pts(4 * eq._M_grid), + fourier_pts(1) / eq.NFP, + ], + NFP=eq.NFP, + ) grid_data = eq.compute( names=Bounce2D.required_names + ["cvdrift", "gbdrift"], grid=grid ) @@ -1558,6 +1557,7 @@ def test_binormal_drift_bounce2d(self): N=N, iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), + N_B=2 * N, alpha=data["alpha"], # - 2 * np.pi * data["iota"], num_transit=4, Bref=data["Bref"], @@ -1581,6 +1581,7 @@ def test_binormal_drift_bounce2d(self): pitch_inv=pitch_inv, num_well=1, check=True, + plot=True, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." From 87dc90d55d4356bc3589c859ec516feb4987b51f Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 21 Sep 2024 20:01:01 -0400 Subject: [PATCH 16/44] Improving API to avoid redundant computation --- desc/integrals/basis.py | 34 +++--- desc/integrals/bounce_integral.py | 121 +++++++++----------- desc/integrals/bounce_utils.py | 43 ++++--- tests/test_integrals.py | 184 +++++++++++++++--------------- 4 files changed, 190 insertions(+), 192 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 956e3ddcc..9dfdbdf05 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -1,4 +1,4 @@ -"""Fast transformable basis.""" +"""Fast transformable series.""" from functools import partial @@ -100,7 +100,7 @@ def _chebcast(cheb, arr): return cheb if jnp.ndim(arr) < cheb.ndim else cheb[jnp.newaxis] -class FourierChebyshevBasis(IOAble): +class FourierChebyshevSeries(IOAble): """Fourier-Chebyshev series. f(x, y) = ∑ₘₙ aₘₙ ψₘ(x) Tₙ(y) @@ -127,7 +127,7 @@ class FourierChebyshevBasis(IOAble): """ def __init__(self, f, domain=(-1, 1), lobatto=False): - """Interpolate Fourier-Chebyshev basis to ``f``. + """Interpolate Fourier-Chebyshev series to ``f``. Parameters ---------- @@ -147,10 +147,10 @@ def __init__(self, f, domain=(-1, 1), lobatto=False): self.domain = tuple(domain) errorif(lobatto, NotImplementedError, "JAX hasn't implemented type 1 DCT.") self.lobatto = bool(lobatto) - self._c = FourierChebyshevBasis._fast_transform(f, self.lobatto) + self._c = FourierChebyshevSeries._transform(f, self.lobatto) @staticmethod - def _fast_transform(f, lobatto): + def _transform(f, lobatto): N = f.shape[-1] return rfft( dct(f, type=2 - lobatto, axis=-1) / (N - lobatto), @@ -223,7 +223,7 @@ def evaluate(self, M, N): return fq def harmonics(self): - """Spectral coefficients aₘₙ of the interpolating polynomial. + """Spectral coefficients aₘₙ of the interpolating trigonometric polynomial. Transform Fourier interpolant harmonics to Nyquist trigonometric interpolant harmonics so that the coefficients are all real. @@ -232,7 +232,7 @@ def harmonics(self): ------- a_mn : jnp.ndarray Shape (..., M, N). - Real valued spectral coefficients for Fourier-Chebyshev basis. + Real valued spectral coefficients for Fourier-Chebyshev series. """ a_mn = harmonic(cheb_from_dct(self._c), self.M, axis=-2) @@ -240,16 +240,16 @@ def harmonics(self): return a_mn def compute_cheb(self, x): - """Evaluate Fourier basis at ``x`` to obtain set of 1D Chebyshev coefficients. + """Evaluate Fourier series at ``x`` to obtain set of 1D Chebyshev coefficients. Parameters ---------- x : jnp.ndarray - Points to evaluate Fourier basis. + Points to evaluate Fourier series. Returns ------- - cheb : ChebyshevBasisSet + cheb : PiecewiseChebyshevSeries Chebyshev coefficients αₙ(x=``x``) for f(x, y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). """ @@ -257,10 +257,10 @@ def compute_cheb(self, x): x = jnp.atleast_1d(x)[..., jnp.newaxis] cheb = cheb_from_dct(irfft_non_uniform(x, self._c, self.M, axis=-2)) assert cheb.shape[-2:] == (x.shape[-2], self.N) - return ChebyshevBasisSet(cheb, self.domain) + return PiecewiseChebyshevSeries(cheb, self.domain) -class ChebyshevBasisSet(IOAble): +class PiecewiseChebyshevSeries(IOAble): """Chebyshev series. { fₓ | fₓ : y ↦ ∑ₙ₌₀ᴺ⁻¹ aₙ(x) Tₙ(y) } @@ -272,16 +272,16 @@ class ChebyshevBasisSet(IOAble): Shape (..., M, N). Chebyshev coefficients αₙ(x) for fₓ(y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). M : int - Number of functions in this basis set. + Number of cuts. N : int - Chebyshev spectral resolution. + Chebyshev spectral resolution of each cut. domain : (float, float) Domain for y coordinates. """ def __init__(self, cheb, domain=(-1, 1)): - """Make Chebyshev series basis from given coefficients. + """Make piecewise series from given Chebyshev coefficients. Parameters ---------- @@ -298,7 +298,7 @@ def __init__(self, cheb, domain=(-1, 1)): @property def M(self): - """Number of functions in this basis set.""" + """Number of cuts.""" return self.cheb.shape[-2] @property @@ -393,7 +393,7 @@ def eval1d(self, z, cheb=None): ------- f : jnp.ndarray Shape z.shape. - Chebyshev basis evaluated at z. + Chebyshev series evaluated at z. """ cheb = _chebcast(setdefault(cheb, self.cheb), z) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 7d91a0960..95a4a27f2 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -6,7 +6,7 @@ from orthax.legendre import leggauss from desc.backend import dct, jnp -from desc.integrals.basis import ChebyshevBasisSet, FourierChebyshevBasis +from desc.integrals.basis import FourierChebyshevSeries, PiecewiseChebyshevSeries from desc.integrals.bounce_utils import ( _bounce_quadrature, _check_bounce_points, @@ -86,7 +86,7 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): Returns ------- - a : FourierChebyshevBasis + a : FourierChebyshevSeries Spectral coefficients of f(α, ζ). """ @@ -95,7 +95,7 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): f = grid.meshgrid_reshape(f, "rtz") M, N = nodes.shape[-3], nodes.shape[-2] - return FourierChebyshevBasis( + return FourierChebyshevSeries( f=interp_rfft2( # Interpolate to nodes in Clebsch space, # which is not a tensor product node set in DESC space. @@ -130,7 +130,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Returns ------- - T, B : (ChebyshevBasisSet, ChebyshevBasisSet) + T, B : (PiecewiseChebyshevSeries, PiecewiseChebyshevSeries) Set of 1D Chebyshev spectral coefficients of |B| along field line. {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the sequence of poloidal coordinates that specify the field line. @@ -142,7 +142,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): # Evaluating set of single variable maps is more efficient than evaluating # multivariable map, so we project θ to a set of Chebyshev series. - T = FourierChebyshevBasis(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) + T = FourierChebyshevSeries(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) theta = T.evaluate(N_B) xq = jnp.stack( [theta, jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape)], axis=-1 @@ -153,7 +153,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): axes=(-1, -2), ).reshape(*alpha.shape, N_B) # Need |B| parameterized by single variable to compute roots. - B = ChebyshevBasisSet(cheb_from_dct(dct(B, type=2, axis=-1)) / N_B, T.domain) + B = PiecewiseChebyshevSeries(cheb_from_dct(dct(B, type=2, axis=-1)) / N_B, T.domain) return T, B @@ -304,16 +304,16 @@ def __init__( grid, data, theta, - N_B, + N_B=32, + num_transit=32, # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb alpha=0.0, - num_transit=32, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), + *, Bref=1.0, Lref=1.0, - *, is_reshaped=False, check=False, **kwargs, @@ -323,7 +323,9 @@ def __init__( Notes ----- Performance may improve significantly if the spectral - resolutions ``m``, ``n``, ``M``, and ``N`` are powers of two. + resolutions ``m``, ``n``, ``M``, ``N``, and ``N_B`` are powers of two. + It is possible for the optimal ``M``, ``N``, and ``N_B`` to decrease + when ``m`` and ``n`` increase, though ``M`` is usually still large. Parameters ---------- @@ -340,7 +342,7 @@ def __init__( DESC coordinates θ sourced from the Clebsch coordinates ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. N_B : int - Desired Chebyshev spectral resolution for |B|. Preferably power of 2. + Desired Chebyshev spectral resolution for |B|. alpha : float Starting field line poloidal label. num_transit : int @@ -452,8 +454,11 @@ def compute_theta(eq, L, M=512, N=16, clebsch=None, **kwargs): """ if clebsch is None: - clebsch = FourierChebyshevBasis.nodes( - check_posint(M), check_posint(N), L, domain=(0, 2 * jnp.pi) + clebsch = FourierChebyshevSeries.nodes( + check_posint(M), + check_posint(N), + check_posint(L), + domain=(0, 2 * jnp.pi), ) return eq.map_coordinates( coords=clebsch, @@ -506,7 +511,7 @@ def points(self, pitch_inv, *, num_well=None): Returns ------- - z1, z2 : (jnp.ndarray, jnp.ndarray) + z1, z2 : tuple[jnp.ndarray] Shape (L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the @@ -519,15 +524,15 @@ def points(self, pitch_inv, *, num_well=None): """ pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. - z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) - return _swap_pl(z1), _swap_pl(z2) + z1, z2 = map(_swap_pl, self._B.intersect1d(pitch_inv, num_intersect=num_well)) + return z1, z2 - def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): + def check_points(self, points, pitch_inv, *, plot=True, **kwargs): """Check that bounce points are computed correctly. Parameters ---------- - z1, z2 : jnp.ndarray + points : tuple[jnp.ndarray] Shape (L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the @@ -541,7 +546,7 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): Whether to plot the field lines and bounce points of the given pitch angles. kwargs : dict Keyword arguments into - ``desc/integrals/basis.py::ChebyshevBasisSet.plot1d``. + ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. Returns ------- @@ -550,8 +555,8 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """ return self._B.check_intersect1d( - z1=_swap_pl(z1), - z2=_swap_pl(z2), + z1=_swap_pl(points[0]), + z2=_swap_pl(points[1]), k=atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T, plot=plot, **_set_default_plot_kwargs(kwargs), @@ -560,11 +565,11 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): def integrate( self, integrand, + points, pitch_inv, f=None, weight=None, *, - num_well=None, check=False, plot=False, ): @@ -580,6 +585,11 @@ def integrate( accept the arrays in ``f`` as arguments as well as the additional keyword arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. + points : tuple[jnp.ndarray] + Shape (L, num_pitch, num_well). + ζ coordinates of bounce points. The points are ordered and grouped such + that the straight line path between ``z1`` and ``z2`` resides in the + epigraph of |B|. pitch_inv : jnp.ndarray Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by @@ -597,16 +607,6 @@ def integrate( the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` interpolated to the deepest point in that magnetic well. Use the method ``Bounce2D.reshape_data`` to reshape the data into the expected shape. - num_well : int or None - Specify to return the first ``num_well`` pairs of bounce points for each - pitch along each field line. This is useful if ``num_well`` tightly - bounds the actual number. As a reference, there are typically 20 wells - per toroidal transit for a given pitch. You can check this by plotting - the field lines with the ``check_points`` method. - - If not specified, then all bounce points are returned. If there were fewer - wells detected along a field line than the size of the last axis of the - returned arrays, then that axis is padded with zero. check : bool Flag for debugging. Must be false for JAX transformations. plot : bool @@ -616,7 +616,7 @@ def integrate( Returns ------- result : jnp.ndarray - Shape (L, num_pitch, num_well). + Shape is same as input points. Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. @@ -626,18 +626,17 @@ def integrate( if not isinstance(f, (list, tuple)): f = [f] if isinstance(f, (jnp.ndarray, np.ndarray)) else list(f) + points = map(_swap_pl, points) pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T - # Expects pitch_inv shape (num_pitch, L) if B.cheb.shape[0] is L. - z1, z2 = self._B.intersect1d(pitch_inv, num_intersect=num_well) - result = self._integrate(z1, z2, integrand, pitch_inv, f, check, plot) + result = self._integrate(integrand, points, pitch_inv, f, check, plot) return result - def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): + def _integrate(self, integrand, points, pitch_inv, f, check, plot): """Bounce integrate ∫ f(λ, ℓ) dℓ. Parameters ---------- - z1, z2 : jnp.ndarray + points : jnp.ndarray Shape (num_pitch, num_well) or (num_pitch, L, num_well). pitch_inv : jnp.ndarray Shape (num_pitch, ) or (num_pitch, L). @@ -645,11 +644,13 @@ def _integrate(self, z1, z2, integrand, pitch_inv, f, check, plot): Shape (m, n) or (L, m, n). """ + z1, z2 = points shape = [*z1.shape, self._x.size] zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) ) + # Expects shape (num_pitch, L) if T.cheb.shape[0] is L. Q = jnp.stack([self._T.eval1d(zeta), zeta], axis=-1) b_sup_z = irfft2_non_uniform( @@ -697,7 +698,7 @@ def plot(self, l, pitch_inv=None, **kwargs): specified by Clebsch coordinate ρ(l) will be plotted. kwargs : dict Keyword arguments into - ``desc/integrals/basis.py::ChebyshevBasisSet.plot1d``. + ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. Returns ------- @@ -707,7 +708,7 @@ def plot(self, l, pitch_inv=None, **kwargs): """ B = self._B if B.cheb.ndim > 2: - B = ChebyshevBasisSet(B.cheb[l], B.domain) + B = PiecewiseChebyshevSeries(B.cheb[l], B.domain) if pitch_inv is not None: errorif( pitch_inv.ndim > 1, @@ -801,9 +802,9 @@ def __init__( data, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), + *, Bref=1.0, Lref=1.0, - *, is_reshaped=False, check=False, **kwargs, @@ -932,7 +933,7 @@ def points(self, pitch_inv, *, num_well=None): Returns ------- - z1, z2 : (jnp.ndarray, jnp.ndarray) + z1, z2 : tuple[jnp.ndarray] Shape (M, L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the @@ -945,12 +946,12 @@ def points(self, pitch_inv, *, num_well=None): """ return bounce_points(pitch_inv, self._zeta, self.B, self._dB_dz, num_well) - def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): + def check_points(self, points, pitch_inv, *, plot=True, **kwargs): """Check that bounce points are computed correctly. Parameters ---------- - z1, z2 : (jnp.ndarray, jnp.ndarray) + points : tuple[jnp.ndarray] Shape (M, L, num_pitch, num_well). ζ coordinates of bounce points. The points are ordered and grouped such that the straight line path between ``z1`` and ``z2`` resides in the @@ -972,8 +973,8 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): """ return _check_bounce_points( - z1=z1, - z2=z2, + z1=points[0], + z2=points[1], pitch_inv=pitch_inv, knots=self._zeta, B=self.B, @@ -984,11 +985,11 @@ def check_points(self, z1, z2, pitch_inv, *, plot=True, **kwargs): def integrate( self, integrand, + points, pitch_inv, f=None, weight=None, *, - num_well=None, method="cubic", batch=True, check=False, @@ -1006,6 +1007,11 @@ def integrate( accept the arrays in ``f`` as arguments as well as the additional keyword arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. + points : tuple[jnp.ndarray] + Shape (M, L, num_pitch, num_well). + ζ coordinates of bounce points. The points are ordered and grouped such + that the straight line path between ``z1`` and ``z2`` resides in the + epigraph of |B|. pitch_inv : jnp.ndarray Shape (M, L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(α,ρ) is specified by @@ -1023,16 +1029,6 @@ def integrate( the returned value is w(j) ∫ f(λ, ℓ) dℓ, where w(j) is ``weight`` interpolated to the deepest point in that magnetic well. Use the method ``Bounce1D.reshape_data`` to reshape the data into the expected shape. - num_well : int or None - Specify to return the first ``num_well`` pairs of bounce points for each - pitch along each field line. This is useful if ``num_well`` tightly - bounds the actual number. As a reference, there are typically 20 wells - per toroidal transit for a given pitch. You can check this by plotting - the field lines with the ``check_points`` method. - - If not specified, then all bounce points are returned. If there were fewer - wells detected along a field line than the size of the last axis of the - returned arrays, then that axis is padded with zero. method : str Method of interpolation. See https://interpax.readthedocs.io/en/latest/_api/interpax.interp1d.html. @@ -1048,18 +1044,16 @@ def integrate( Returns ------- result : jnp.ndarray - Shape (M, L, num_pitch, num_well). + Shape is same as input points. Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. """ - z1, z2 = self.points(pitch_inv, num_well=num_well) result = _bounce_quadrature( x=self._x, w=self._w, - z1=z1, - z2=z2, integrand=integrand, + points=points, pitch_inv=pitch_inv, f=setdefault(f, []), data=self._data, @@ -1072,14 +1066,13 @@ def integrate( if weight is not None: result *= interp_to_argmin( weight, - z1, - z2, + points, self._zeta, self.B, self._dB_dz, method, ) - assert result.shape == z1.shape + assert result.shape == points[0].shape return result def plot(self, m, l, pitch_inv=None, **kwargs): diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 27ca8d05e..57eb88a38 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -55,7 +55,7 @@ def get_alpha(alpha_0, iota, num_transit, period): return alpha -def get_pitch_inv(min_B, max_B, num, relative_shift=1e-6): +def get_pitch_inv(min_B, max_B, num_pitch, relative_shift=1e-6): """Return 1/λ values for quadrature between ``min_B`` and ``max_B``. Parameters @@ -64,7 +64,7 @@ def get_pitch_inv(min_B, max_B, num, relative_shift=1e-6): Minimum |B| value. max_B : jnp.ndarray Maximum |B| value. - num : int + num_pitch : int Number of values, not including endpoints. relative_shift : float Relative amount to shift maxima down and minima up to avoid floating point @@ -82,8 +82,10 @@ def get_pitch_inv(min_B, max_B, num, relative_shift=1e-6): min_B = (1 + relative_shift) * min_B max_B = (1 - relative_shift) * max_B # Samples should be uniformly spaced in |B| and not λ (GitHub issue #1228). - pitch_inv = jnp.moveaxis(composite_linspace(jnp.stack([min_B, max_B]), num), 0, -1) - assert pitch_inv.shape == (*min_B.shape, num + 2) + pitch_inv = jnp.moveaxis( + composite_linspace(jnp.stack([min_B, max_B]), num_pitch), 0, -1 + ) + assert pitch_inv.shape == (*min_B.shape, num_pitch + 2) return pitch_inv @@ -310,13 +312,13 @@ def _check_bounce_points(z1, z2, pitch_inv, knots, B, plot=True, **kwargs): def _bounce_quadrature( x, w, - z1, - z2, integrand, + points, pitch_inv, f, data, knots, + *, method="cubic", batch=True, check=False, @@ -332,17 +334,17 @@ def _bounce_quadrature( w : jnp.ndarray Shape (num_quad, ). Quadrature weights. - z1, z2 : jnp.ndarray - Shape (..., num_pitch, num_well). - ζ coordinates of bounce points. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in the - epigraph of |B|. integrand : callable The composition operator on the set of functions in ``f`` that maps the functions in ``f`` to the integrand f(λ, ℓ) in ∫ f(λ, ℓ) dℓ. It should accept the arrays in ``f`` as arguments as well as the additional keyword arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. + points : jnp.ndarray + Shape (..., num_pitch, num_well). + ζ coordinates of bounce points. The points are ordered and grouped such + that the straight line path between ``z1`` and ``z2`` resides in the + epigraph of |B|. pitch_inv : jnp.ndarray Shape (..., num_pitch). 1/λ values to compute the bounce integrals. @@ -379,6 +381,7 @@ def _bounce_quadrature( """ errorif(x.ndim != 1 or x.shape != w.shape) + z1, z2 = points errorif(z1.ndim < 2 or z1.shape != z2.shape) pitch_inv = jnp.atleast_1d(pitch_inv) if not isinstance(f, (list, tuple)): @@ -389,8 +392,8 @@ def _bounce_quadrature( result = _interpolate_and_integrate( w=w, Q=bijection_from_disc(x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]), - pitch_inv=pitch_inv, integrand=integrand, + pitch_inv=pitch_inv, f=f, data=data, knots=knots, @@ -406,8 +409,8 @@ def loop(z): # over num well axis return None, _interpolate_and_integrate( w=w, Q=bijection_from_disc(x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]), - pitch_inv=pitch_inv, integrand=integrand, + pitch_inv=pitch_inv, f=f, data=data, knots=knots, @@ -429,8 +432,8 @@ def loop(z): # over num well axis def _interpolate_and_integrate( w, Q, - pitch_inv, integrand, + pitch_inv, f, data, knots, @@ -615,7 +618,7 @@ def _where_for_argmin(z1, z2, ext, g_ext, upper_sentinel): def interp_to_argmin( - h, z1, z2, knots, g, dg_dz, method="cubic", beta=-100, upper_sentinel=1e2 + h, points, knots, g, dg_dz, method="cubic", beta=-100, upper_sentinel=1e2 ): """Interpolate ``h`` to the deepest point of ``g`` between ``z1`` and ``z2``. @@ -626,9 +629,10 @@ def interp_to_argmin( h : jnp.ndarray Shape (..., N). Values evaluated on ``knots`` to interpolate. - z1, z2 : jnp.ndarray + points : jnp.ndarray Shape (..., num_pitch, W). Boundaries to detect argmin between. + First (second) element stores left (right) boundaries. knots : jnp.ndarray Shape (N, ). z coordinates of spline knots. Must be strictly increasing. @@ -667,6 +671,7 @@ def interp_to_argmin( Shape (..., num_pitch, W). """ + z1, z2 = points assert z1.ndim == z2.ndim >= 2 and z1.shape == z2.shape ext, g_ext = _get_extrema(knots, g, dg_dz, sentinel=0) # Our softargmax(x) does the proper shift to compute softargmax(x - max(x)), @@ -684,7 +689,7 @@ def interp_to_argmin( return h -def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): +def interp_to_argmin_hard(h, points, knots, g, dg_dz, method="cubic"): """Interpolate ``h`` to the deepest point of ``g`` between ``z1`` and ``z2``. Let E = {ζ ∣ ζ₁ < ζ < ζ₂} and A ∈ argmin_E g(ζ). Returns h(A). @@ -700,9 +705,10 @@ def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): h : jnp.ndarray Shape (..., N). Values evaluated on ``knots`` to interpolate. - z1, z2 : jnp.ndarray + points : jnp.ndarray Shape (..., num_pitch, W). Boundaries to detect argmin between. + First (second) element stores left (right) boundaries. knots : jnp.ndarray Shape (N, ). z coordinates of spline knots. Must be strictly increasing. @@ -727,6 +733,7 @@ def interp_to_argmin_hard(h, z1, z2, knots, g, dg_dz, method="cubic"): Shape (..., num_pitch, W). """ + z1, z2 = points assert z1.ndim == z2.ndim >= 2 and z1.shape == z2.shape ext, g_ext = _get_extrema(knots, g, dg_dz, sentinel=0) # We can use the non-differentiable max because we actually want the gradients diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 9cfeeda29..379ca412b 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -34,7 +34,7 @@ surface_variance, virtual_casing_biot_savart, ) -from desc.integrals.basis import FourierChebyshevBasis +from desc.integrals.basis import FourierChebyshevSeries from desc.integrals.bounce_utils import ( _get_extrema, bounce_points, @@ -724,9 +724,17 @@ def test_biest_interpolators(self): np.testing.assert_allclose(g1, ff) -class TestBounce1DPoints: +class TestBouncePoints: """Test that bounce points are computed correctly.""" + @staticmethod + def _cheb_intersect(cheb, k): + cheb = cheb.copy() + cheb[0] = cheb[0] - k + roots = chebroots(cheb) + intersect = roots[np.logical_and(np.isreal(roots), np.abs(roots.real) < 1)].real + return intersect + @staticmethod def filter(z1, z2): """Remove bounce points whose integrals have zero measure.""" @@ -745,7 +753,7 @@ def test_z1_first(self): z1, z2 = bounce_points( pitch_inv, knots, B.c.T, B.derivative().c.T, check=True, include_knots=True ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size np.testing.assert_allclose(z1, intersect[0::2]) np.testing.assert_allclose(z2, intersect[1::2]) @@ -762,7 +770,7 @@ def test_z2_first(self): z1, z2 = bounce_points( pitch_inv, k, B.c.T, B.derivative().c.T, check=True, include_knots=True ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size np.testing.assert_allclose(z1, intersect[1:-1:2]) np.testing.assert_allclose(z2, intersect[0::2][1:]) @@ -783,7 +791,7 @@ def test_z1_before_extrema(self): z1, z2 = bounce_points( pitch_inv, k, B.c.T, dB_dz.c.T, check=True, include_knots=True ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size intersect = B.solve(pitch_inv, extrapolate=False) np.testing.assert_allclose(z1[1], 1.982767, rtol=1e-6) @@ -810,7 +818,7 @@ def test_z2_before_extrema(self): z1, z2 = bounce_points( pitch_inv, k, B.c.T, dB_dz.c.T, check=True, include_knots=True ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size intersect = B.solve(pitch_inv, extrapolate=False) np.testing.assert_allclose(z1, intersect[[0, -2]]) @@ -839,7 +847,7 @@ def test_extrema_first_and_before_z1(self): start=k[2], include_knots=True, ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size intersect = B.solve(pitch_inv, extrapolate=False) np.testing.assert_allclose(z1[0], 0.835319, rtol=1e-6) @@ -864,7 +872,7 @@ def test_extrema_first_and_before_z2(self): z1, z2 = bounce_points( pitch_inv, k, B.c.T, dB_dz.c.T, check=True, include_knots=True ) - z1, z2 = TestBounce1DPoints.filter(z1, z2) + z1, z2 = TestBouncePoints.filter(z1, z2) assert z1.size and z2.size # Our routine correctly detects intersection, while scipy, jnp.root fails. intersect = B.solve(pitch_inv, extrapolate=False) @@ -895,6 +903,27 @@ def test_get_extrema(self): np.testing.assert_allclose(ext[idx], ext_scipy) np.testing.assert_allclose(B_ext[idx], B_ext_scipy) + @pytest.mark.unit + def test_z1_first_chebyshev(self): + """Test that bounce points are computed correctly.""" + + def f(z): + return -2 * np.cos(1 / (0.1 + z**2)) + 2 + + M, N = 1, 10 + alpha, zeta = FourierChebyshevSeries.nodes(M, N).T + cheb = FourierChebyshevSeries(f(zeta).reshape(M, N)).compute_cheb( + fourier_pts(M) + ) + pitch_inv = 3 + z1, z2 = cheb.intersect1d(pitch_inv) + cheb.check_intersect1d(z1, z2, pitch_inv) + z1, z2 = TestBouncePoints.filter(z1, z2) + + r = self._cheb_intersect(chebinterpolate(f, N - 1), pitch_inv) + np.testing.assert_allclose(z1, r[np.isclose(r, -0.24, atol=1e-1)]) + np.testing.assert_allclose(z2, r[np.isclose(r, 0.24, atol=1e-1)]) + def _mod_cheb_gauss(deg): x, w = chebgauss(deg) @@ -908,7 +937,7 @@ def _mod_chebu_gauss(deg): return x, w -class TestBounce1DQuadrature: +class TestBounceQuadrature: """Test bounce quadrature.""" @pytest.mark.unit @@ -956,7 +985,10 @@ def test_bounce_quadrature(self, is_strong, quad, automorphism): check=True, **kwargs, ) - result = bounce.integrate(integrand, pitch_inv, check=True, plot=True) + points = bounce.points(pitch_inv, num_well=1) + np.testing.assert_allclose(points[0], z1) + np.testing.assert_allclose(points[1], z2) + result = bounce.integrate(integrand, points, pitch_inv, check=True, plot=True) assert np.count_nonzero(result) == 1 np.testing.assert_allclose(result.sum(), truth, rtol=1e-4) @@ -995,14 +1027,14 @@ def elliptic_incomplete(k2): # Scipy's elliptic integrals are broken. # https://github.com/scipy/scipy/issues/20525. k = np.sqrt(k2) - K = TestBounce1DQuadrature._adaptive_elliptic(K_integrand, k) - E = TestBounce1DQuadrature._adaptive_elliptic(E_integrand, k) + K = TestBounceQuadrature._adaptive_elliptic(K_integrand, k) + E = TestBounceQuadrature._adaptive_elliptic(E_integrand, k) # Make sure scipy's adaptive quadrature is not broken. np.testing.assert_allclose( - K, TestBounce1DQuadrature._fixed_elliptic(K_integrand, k, 10) + K, TestBounceQuadrature._fixed_elliptic(K_integrand, k, 10) ) np.testing.assert_allclose( - E, TestBounce1DQuadrature._fixed_elliptic(E_integrand, k, 10) + E, TestBounceQuadrature._fixed_elliptic(E_integrand, k, 10) ) I_0 = 4 / k * K @@ -1016,32 +1048,32 @@ def elliptic_incomplete(k2): # Check for math mistakes. np.testing.assert_allclose( I_2, - TestBounce1DQuadrature._adaptive_elliptic( + TestBounceQuadrature._adaptive_elliptic( lambda Z, k: 2 / np.sqrt(k**2 - np.sin(Z / 2) ** 2) * Z * np.sin(Z), k ), ) np.testing.assert_allclose( I_3, - TestBounce1DQuadrature._adaptive_elliptic( + TestBounceQuadrature._adaptive_elliptic( lambda Z, k: 2 * np.sqrt(k**2 - np.sin(Z / 2) ** 2) * Z * np.sin(Z), k ), ) np.testing.assert_allclose( I_4, - TestBounce1DQuadrature._adaptive_elliptic( + TestBounceQuadrature._adaptive_elliptic( lambda Z, k: 2 / np.sqrt(k**2 - np.sin(Z / 2) ** 2) * np.sin(Z) ** 2, k ), ) np.testing.assert_allclose( I_5, - TestBounce1DQuadrature._adaptive_elliptic( + TestBounceQuadrature._adaptive_elliptic( lambda Z, k: 2 * np.sqrt(k**2 - np.sin(Z / 2) ** 2) * np.sin(Z) ** 2, k ), ) # scipy fails np.testing.assert_allclose( I_6, - TestBounce1DQuadrature._fixed_elliptic( + TestBounceQuadrature._fixed_elliptic( lambda Z, k: 2 / np.sqrt(k**2 - np.sin(Z / 2) ** 2) * np.cos(Z), k, deg=11, @@ -1049,7 +1081,7 @@ def elliptic_incomplete(k2): ) np.testing.assert_allclose( I_7, - TestBounce1DQuadrature._adaptive_elliptic( + TestBounceQuadrature._adaptive_elliptic( lambda Z, k: 2 * np.sqrt(k**2 - np.sin(Z / 2) ** 2) * np.cos(Z), k ), ) @@ -1094,23 +1126,27 @@ def test_bounce1d_checks(self): Bounce1D.required_names + ["min_tz |B|", "max_tz |B|", "g_zz"], grid=grid ) # 5. Make the bounce integration operator. - bounce = Bounce1D( - grid.source_grid, - data, - quad=leggauss(3), # not checking quadrature accuracy in this test - check=True, - ) + bounce = Bounce1D(grid.source_grid, data, quad=leggauss(3), check=True) pitch_inv = bounce.get_pitch_inv( - grid.compress(data["min_tz |B|"]), grid.compress(data["max_tz |B|"]), 10 - ) + min_B=grid.compress(data["min_tz |B|"]), + max_B=grid.compress(data["max_tz |B|"]), + num_pitch=10, + ) + # 6. Compute bounce points. + points = bounce.points(pitch_inv) + # 7. Optionally check for correctness of bounce points. + bounce.check_points(points, pitch_inv, plot=False) + # 8. Integrate. num = bounce.integrate( integrand=TestBounce1D._example_numerator, + points=points, pitch_inv=pitch_inv, f=Bounce1D.reshape_data(grid.source_grid, data["g_zz"]), check=True, ) den = bounce.integrate( integrand=TestBounce1D._example_denominator, + points=points, pitch_inv=pitch_inv, check=True, batch=False, @@ -1118,7 +1154,7 @@ def test_bounce1d_checks(self): avg = safediv(num, den) assert np.isfinite(avg).all() and np.count_nonzero(avg) - # 6. Basic manipulation of the output. + # 9. Example manipulation of the output # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) # Group the result by pitch and flux surface. @@ -1132,10 +1168,7 @@ def test_bounce1d_checks(self): nodes = grid.source_grid.meshgrid_reshape(grid.source_grid.nodes[:, :2], "arz") print("(α, ρ):", nodes[m, l, 0]) - # 7. Optionally check for correctness of bounce points - bounce.check_points(*bounce.points(pitch_inv), pitch_inv, plot=False) - - # 8. Plotting + # 10. Plotting fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=False) return fig @@ -1169,19 +1202,11 @@ def dg_dz(z): "|B|_z|r,a": dg_dz(zeta), }, ) - z1 = np.array(0, ndmin=4) - z2 = np.array(2 * np.pi, ndmin=4) + points = (np.array(0, ndmin=4), np.array(2 * np.pi, ndmin=4)) argmin = 5.61719 h_min = h(argmin) - result = func( - h=h(zeta), - z1=z1, - z2=z2, - knots=zeta, - g=bounce.B, - dg_dz=bounce._dB_dz, - ) - assert result.shape == z1.shape + result = func(h(zeta), points, zeta, bounce.B, bounce._dB_dz) + assert result.shape == points[0].shape np.testing.assert_allclose(h_min, result, rtol=1e-3) @staticmethod @@ -1313,7 +1338,7 @@ def drift_analytic(data): pitch_inv = get_pitch_inv(np.min(B), np.max(B), 100)[:-1] k2 = 0.5 * ((1 - B0 / pitch_inv) / (epsilon * B0 / pitch_inv) + 1) I_0, I_1, I_2, I_3, I_4, I_5, I_6, I_7 = ( - TestBounce1DQuadrature.elliptic_incomplete(k2) + TestBounceQuadrature.elliptic_incomplete(k2) ) y = np.sqrt(2 * epsilon * B0 / pitch_inv) I_0, I_2, I_4, I_6 = map(lambda I: I / y, (I_0, I_2, I_4, I_6)) @@ -1360,21 +1385,21 @@ def test_binormal_drift_bounce1d(self): Lref=data["a"], check=True, ) - bounce.check_points(*bounce.points(pitch_inv), pitch_inv, plot=False) + points = bounce.points(pitch_inv, num_well=1) + bounce.check_points(points, pitch_inv, plot=False) f = Bounce1D.reshape_data(things["grid"].source_grid, cvdrift, gbdrift) drift_numerical_num = bounce.integrate( integrand=TestBounce1D.drift_num_integrand, + points=points, pitch_inv=pitch_inv, f=f, - num_well=1, check=True, - # plot=True, # noqa: E800 ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, + points=points, pitch_inv=pitch_inv, - num_well=1, weight=np.ones(data["zeta"].size), check=True, ) @@ -1446,15 +1471,21 @@ def integrand_grad(*args, **kwargs2): return grad_fun(*args, *kwargs2.values()) def fun1(pitch): - return bounce.integrate(integrand, 1 / pitch, check=False, **kwargs).sum() + return bounce.integrate( + integrand, bounce.points(1 / pitch), 1 / pitch, check=False, **kwargs + ).sum() def fun2(pitch): return bounce.integrate( - integrand_grad, 1 / pitch, check=True, **kwargs + integrand_grad, + bounce.points(1 / pitch), + 1 / pitch, + check=True, + **kwargs, ).sum() pitch = 1.0 - # can easily obtain from math or just extrapolate from analytic expression plot + # can obtain from math or just extrapolate from analytic expression plot analytic_approximation_of_gradient = 650 np.testing.assert_allclose( grad(fun1)(pitch), analytic_approximation_of_gradient, rtol=1e-3 @@ -1465,37 +1496,6 @@ def fun2(pitch): np.testing.assert_allclose(fun2(pitch), -171500, rtol=1e-1) -class TestBounce2DPoints: - """Test that bounce points are computed correctly.""" - - @staticmethod - def _cheb_intersect(cheb, k): - cheb = cheb.copy() - cheb[0] = cheb[0] - k - roots = chebroots(cheb) - intersect = roots[np.logical_and(np.isreal(roots), np.abs(roots.real) < 1)].real - return intersect - - @pytest.mark.unit - def test_z1_first(self): - """Test that bounce points are computed correctly.""" - - def f(z): - return -2 * np.cos(1 / (0.1 + z**2)) + 2 - - M, N = 1, 10 - alpha, zeta = FourierChebyshevBasis.nodes(M, N).T - cheb = FourierChebyshevBasis(f(zeta).reshape(M, N)).compute_cheb(fourier_pts(M)) - pitch_inv = 3 - z1, z2 = cheb.intersect1d(pitch_inv) - cheb.check_intersect1d(z1, z2, pitch_inv) - z1, z2 = TestBounce1DPoints.filter(z1, z2) - - r = self._cheb_intersect(chebinterpolate(f, N - 1), pitch_inv) - np.testing.assert_allclose(z1, r[np.isclose(r, -0.24, atol=1e-1)]) - np.testing.assert_allclose(z2, r[np.isclose(r, 0.24, atol=1e-1)]) - - class TestBounce2D: """Test bounce integration with two-dimensional pseudo-spectral methods.""" @@ -1508,7 +1508,8 @@ class TestBounce2D: ], ) def test_alpha_sequence(self, alpha_0, iota, num_period, period): - """Test field line labels.""" + """Test field line label is updated correctly after toroidal transits.""" + # this test will be more useful once zeta != phi iota = np.atleast_1d(iota) alphas = get_alpha(alpha_0, iota, num_period, period) assert alphas.shape == (iota.size, num_period) @@ -1527,11 +1528,7 @@ def test_binormal_drift_bounce2d(self): # FIXME: Change LinearGrid to default to Fourier points nodes. Interpolation # fails on LinearGrid. grid = Grid.create_meshgrid( - [ - data["rho"], - fourier_pts(4 * eq._M_grid), - fourier_pts(1) / eq.NFP, - ], + [data["rho"], fourier_pts(4 * eq._M_grid), fourier_pts(1) / eq.NFP], NFP=eq.NFP, ) grid_data = eq.compute( @@ -1558,28 +1555,29 @@ def test_binormal_drift_bounce2d(self): iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), N_B=2 * N, - alpha=data["alpha"], # - 2 * np.pi * data["iota"], num_transit=4, + alpha=data["alpha"], # - 2 * np.pi * data["iota"], Bref=data["Bref"], Lref=data["a"], check=True, plot=True, ) - bounce.check_points(*bounce.points(pitch_inv), pitch_inv, plot=True) + points = bounce.points(pitch_inv, num_well=1) + bounce.check_points(points, pitch_inv, plot=True) f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) drift_numerical_num = bounce.integrate( integrand=TestBounce1D.drift_num_integrand, + points=points, pitch_inv=pitch_inv, f=f, - num_well=1, check=True, plot=True, ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, + points=points, pitch_inv=pitch_inv, - num_well=1, check=True, plot=True, ) From 7977695c94f5180c480163ba7c8c30d7e9113e92 Mon Sep 17 00:00:00 2001 From: unalmis Date: Sun, 22 Sep 2024 13:49:17 -0400 Subject: [PATCH 17/44] Document choice for quadrature --- desc/integrals/bounce_integral.py | 15 ++++++++++++--- 1 file changed, 12 insertions(+), 3 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 2cb0bade8..4117ca036 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -272,8 +272,12 @@ class Bounce2D(IOAble): After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. - Fast fourier transforms interpolate functions in the integrand to the - quadrature nodes. + Fast fourier transforms interpolate smooth functions in the integrand to the + quadrature nodes. Quadrature is chosen over Runge-Kutta methods of the form + ∂Fᵢ/∂ζ =f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 + because a fourth order Runge-Kutta method is equivalent to a quadrature + with Simpson's rule. Our quadratures resolve these integrals more + efficiently, and the fixed nature of quadrature performs better on GPUs. Fast transforms are used where possible. Fast multipoint methods are not implemented. For non-uniform interpolation, MMTs are used. It should be @@ -768,7 +772,12 @@ class Bounce1D(IOAble): After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. - Local splines interpolate functions in the integrand to the quadrature nodes. + Local splines interpolate smooth functions in the integrand to the quadrature + nodes. Quadrature is chosen over Runge-Kutta methods of the form + ∂Fᵢ/∂ζ =f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 + because a fourth order Runge-Kutta method is equivalent to a quadrature + with Simpson's rule. Our quadratures resolve these integrals more + efficiently, and the fixed nature of quadrature performs better on GPUs. See Also -------- From 53ca8343c8216441f788a56c5259cd7260d37b5a Mon Sep 17 00:00:00 2001 From: unalmis Date: Sun, 22 Sep 2024 16:59:10 -0400 Subject: [PATCH 18/44] More documentation --- desc/integrals/bounce_integral.py | 53 +++++++++++++++++++------------ tests/test_integrals.py | 9 +++--- 2 files changed, 37 insertions(+), 25 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 4117ca036..0221169bc 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -130,11 +130,10 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Returns ------- - T, B : (PiecewiseChebyshevSeries, PiecewiseChebyshevSeries) - Set of 1D Chebyshev spectral coefficients of |B| along field line. - {|B|_α : ζ ↦ |B|(α, ζ) | α ∈ A } where A = (α₀, α₁, …, αₘ₋₁) is the - sequence of poloidal coordinates that specify the field line. - Likewise with θ. + T, B : tuple[PiecewiseChebyshevSeries] + Set of 1D Chebyshev spectral coefficients of θ along field line. + {θ_α : ζ ↦ θ(α, ζ) | α ∈ A} where A = (α₀, α₁, …, αₘ₋₁) is ``alpha``. + Likewise with |B|. """ if not is_reshaped: @@ -193,7 +192,8 @@ class Bounce2D(IOAble): Magnetic field line with label α, defined by B = ∇ρ × ∇α, is determined from α : ρ, θ, ζ ↦ θ + λ(ρ,θ,ζ) − ι(ρ) [ζ + ω(ρ,θ,ζ)] Interpolate Fourier-Chebyshev series to DESC poloidal coordinate. - θ : α, ζ ↦ tₘₙ exp(jmα) Tₙ(ζ) + θ : α, ζ ↦ tₘₙ exp(jmα) Tₙ(ζ) + Compute |B| along field lines. |B| : α, ζ ↦ bₙ(θ(α, ζ)) Tₙ(ζ) Compute bounce points. r(ζₖ) = |B|(ζₖ) − 1/λ = 0 @@ -249,8 +249,9 @@ class Bounce2D(IOAble): periodic coordinates, e.g. (ϑ, ϕ), to a low order polynomial basis in non-periodic coordinates. For example, one can obtain series coefficients in (α, ϕ) coordinates from those in (ϑ, ϕ) as follows - g : ϑ, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mϑ + nϕ]) - g : α, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mα + (m ι + n)ϕ]) + g : ϑ, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mϑ + nϕ]) + + g : α, ϕ ↦ ∑ₘₙ aₘₙ exp(j [mα + (m ι + n)ϕ]) However, the basis for the latter are trigonometric functions with irrational frequencies, courtesy of the irrational rotational transform. Globally convergent root-finding schemes for that basis (at fixed α) are @@ -274,7 +275,7 @@ class Bounce2D(IOAble): By default, this is a Gauss quadrature after removing the singularity. Fast fourier transforms interpolate smooth functions in the integrand to the quadrature nodes. Quadrature is chosen over Runge-Kutta methods of the form - ∂Fᵢ/∂ζ =f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 + ∂Fᵢ/∂ζ = f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 because a fourth order Runge-Kutta method is equivalent to a quadrature with Simpson's rule. Our quadratures resolve these integrals more efficiently, and the fixed nature of quadrature performs better on GPUs. @@ -289,9 +290,18 @@ class Bounce2D(IOAble): -------- Bounce1D Uses one-dimensional local spline methods for the same task. - An advantage of ``Bounce2D`` over ``Bounce1D`` is that the coordinates on - which the root-finding must be done to map from DESC to Clebsch coords is - fixed to ``L*M*N``, independent of the number of toroidal transits. + + Below are some advantages of ``Bounce2D`` over ``Bounce1D``. + The coordinates on which the root-finding must be done to map from DESC + to Clebsch coords is fixed to ``L*M*N``, independent of the number of + toroidal transits; generating the same data with DESC for input to + ``Bounce1D`` requires ``L*M*N*num_transit``. Furthermore, pseudo-spectral + interpolation of smooth functions such as |B| on each flux surface is more + efficient. This reduces the number of bounce integrals to be done by an + order of magnitude. Furthermore, C1 cubic spline interpolation is + inefficient to reconstruct smooth local maxima of |B|, which might be + important for (strongly) singular bounce integrals whose estimation + depends on ∂|B|/∂ζ there. Attributes ---------- @@ -351,19 +361,20 @@ def __init__( Starting field line poloidal label. num_transit : int Number of toroidal transits to follow field line. - quad : (jnp.ndarray, jnp.ndarray) + quad : tuple[jnp.ndarray] Quadrature points xₖ and weights wₖ for the approximate evaluation of an integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). Default is 32 points. For weak singular integrals, use ``chebgauss2`` from ``desc.integrals.quad_utils``. For strong singular integrals, use ``leggauss``. - automorphism : (Callable, Callable) or None + automorphism : tuple[Callable] or None The first callable should be an automorphism of the real interval [-1, 1]. The second callable should be the derivative of the first. This map defines a change of variable for the bounce integral. The choice made for the automorphism will affect the performance of the quadrature method. For weak singular integrals, use ``None``. - For strong singular integrals, use ``automorphism_sin`` from + For strong singular integrals, use + ``(automorphism_sin,grad_automorphism_sin)`` from ``desc.integrals.quad_utils``. Bref : float Optional. Reference magnetic field strength for normalization. @@ -774,14 +785,15 @@ class Bounce1D(IOAble): By default, this is a Gauss quadrature after removing the singularity. Local splines interpolate smooth functions in the integrand to the quadrature nodes. Quadrature is chosen over Runge-Kutta methods of the form - ∂Fᵢ/∂ζ =f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 + ∂Fᵢ/∂ζ = f(λ,ζ,{Gⱼ}) subject to Fᵢ(ζ₁) = 0 because a fourth order Runge-Kutta method is equivalent to a quadrature with Simpson's rule. Our quadratures resolve these integrals more efficiently, and the fixed nature of quadrature performs better on GPUs. See Also -------- - Bounce2D : Uses two-dimensional pseudo-spectral techniques for the same task. + Bounce2D + Uses two-dimensional pseudo-spectral techniques for the same task. Examples -------- @@ -832,19 +844,20 @@ def __init__( data : dict[str, jnp.ndarray] Data evaluated on ``grid``. Must include names in ``Bounce1D.required_names``. - quad : (jnp.ndarray, jnp.ndarray) + quad : tuple[jnp.ndarray] Quadrature points xₖ and weights wₖ for the approximate evaluation of an integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). Default is 32 points. For weak singular integrals, use ``chebgauss2`` from ``desc.integrals.quad_utils``. For strong singular integrals, use ``leggauss``. - automorphism : (Callable, Callable) or None + automorphism : tuple[Callable] or None The first callable should be an automorphism of the real interval [-1, 1]. The second callable should be the derivative of the first. This map defines a change of variable for the bounce integral. The choice made for the automorphism will affect the performance of the quadrature method. For weak singular integrals, use ``None``. - For strong singular integrals, use ``automorphism_sin`` from + For strong singular integrals, use + ``(automorphism_sin,grad_automorphism_sin)`` from ``desc.integrals.quad_utils``. Bref : float Optional. Reference magnetic field strength for normalization. diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 7142febba..9a8403aca 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -997,10 +997,10 @@ def test_bounce_quadrature(self, is_strong, quad, automorphism): integrand = lambda B, pitch: jnp.sqrt(1 - m * pitch * B) truth = v * 2 * ellipe(m) bounce = Bounce1D( - Grid.create_meshgrid([1, 0, knots], coordinates="raz"), - data, - quad, - automorphism, + grid=Grid.create_meshgrid([1, 0, knots], coordinates="raz"), + data=data, + quad=quad, + automorphism=automorphism, check=True, ) points = bounce.points(pitch_inv, num_well=1) @@ -1557,7 +1557,6 @@ def test_binormal_drift_bounce2d(self): # Compute numerical result. M, N = 512, 16 - # Neo does 200 x 200 for their algorithm, so I suppose this isn't too weird. # I have concern that numpy computes Nyquist frequency component incorrectly, # and the reason high Fourier resolution is required is that with large sample # frequency the incorrect max frequency component is higher than the max From 679697fefee30a57879a1f88812fe861bb8e4096 Mon Sep 17 00:00:00 2001 From: unalmis Date: Tue, 24 Sep 2024 04:29:13 -0400 Subject: [PATCH 19/44] Document multivaluedness bug and how to fix. likely will be done in new PR --- desc/grid.py | 4 ++- desc/integrals/basis.py | 5 +-- desc/integrals/bounce_integral.py | 60 +++++++++++++++++++------------ desc/integrals/bounce_utils.py | 2 ++ desc/integrals/interp_utils.py | 7 ++-- desc/integrals/quad_utils.py | 2 +- tests/test_integrals.py | 32 ++++++++++------- 7 files changed, 68 insertions(+), 44 deletions(-) diff --git a/desc/grid.py b/desc/grid.py index 9546dd733..d701022c9 100644 --- a/desc/grid.py +++ b/desc/grid.py @@ -1037,8 +1037,10 @@ def _create_nodes( # noqa: C901 """ self._NFP = check_posint(NFP, "NFP", False) self._period = (np.inf, 2 * np.pi, 2 * np.pi / self._NFP) - # TODO: + # FIXME: # https://github.com/PlasmaControl/DESC/pull/1204#pullrequestreview-2246771337 + # Quantities like alpha, grad(alpha), etc. are computed incorrectly at + # phi > 2pi / NFP and theta > 2pi / NFP. axis = bool(axis) endpoint = bool(endpoint) theta_period = self.period[1] diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 9dfdbdf05..eb47860e6 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -133,7 +133,7 @@ def __init__(self, f, domain=(-1, 1), lobatto=False): ---------- f : jnp.ndarray Shape (..., M, N). - Samples of real function on the ``FourierChebyshevBasis.nodes`` grid. + Samples of real function on the ``FourierChebyshevSeries.nodes`` grid. domain : (float, float) Domain for y coordinates. Default is [-1, 1]. lobatto : bool @@ -211,7 +211,7 @@ def evaluate(self, M, N): fq : jnp.ndarray Shape (..., M, N) Fourier-Chebyshev series evaluated at - ``FourierChebyshevBasis.nodes(M,N,L,self.domain,self.lobatto)``. + ``FourierChebyshevSeries.nodes(M,N,L,self.domain,self.lobatto)``. """ fq = idct( @@ -510,6 +510,7 @@ def intersect1d(self, k=0.0, *, num_intersect=None, pad_value=0.0): # ``_eps`` from ``domain[-1]``. Edit: For differentiability, we cannot # consider intersects at boundary of Chebyshev polynomial. Again, cases # where this would be incorrect have measure zero. + # TODO: (kaya) look into that plot with M=512, N=32 is_z1 = (df_dy_sign <= 0) & is_intersect is_z2 = (df_dy_sign >= 0) & _in_epigraph_and(is_intersect, df_dy_sign) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 0221169bc..8f95c0b43 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -80,7 +80,7 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): nodes : jnp.ndarray Shape (L, M, N, 2) or (M, N, 2). DESC coordinates (θ, ζ) sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,domain=(0,2*jnp.pi))``. + ``FourierChebyshevSeries.nodes(M,N,domain=(0,2*jnp.pi))``. f : jnp.ndarray Function evaluated on ``grid``. @@ -122,7 +122,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): theta : jnp.ndarray Shape (L, M, N) or (M, N). DESC coordinates θ sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,domain=(0,2*jnp.pi))``. + ``FourierChebyshevSeries.nodes(M,N,domain=(0,2*jnp.pi))``. B : jnp.ndarray |B| evaluated on ``grid``. N_B : int @@ -354,7 +354,7 @@ def __init__( theta : jnp.ndarray Shape (L, M, N). DESC coordinates θ sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + ``FourierChebyshevSeries.nodes(M,N,L,domain=(0,2*jnp.pi))``. N_B : int Desired Chebyshev spectral resolution for |B|. alpha : float @@ -400,6 +400,7 @@ def __init__( check and kwargs.pop("warn", True) and jnp.any(data["B^zeta"] <= 0), msg="(∂ℓ/∂ζ)|ρ,a > 0 is required. Enforcing positive B^ζ.", ) + self._alpha = alpha self._m = grid.num_theta self._n = grid.num_zeta self._x, self._w = get_quadrature(quad, automorphism) @@ -432,9 +433,12 @@ def __init__( # for near omnigenous configurations in particular, (∂|B|/∂α)|ρ,ζ vanishes. # However, the Fourier series of θ(α, ζ) converges slower than desired. # Small discontinuities of quantities evaluated between adjacent cuts of a - # field line vanish at M > 256. + # field line vanish at M > 256. Memory seems much better than DESC's transforms + # though, and we avoid creation of a 1000 intermediate dependency arrays of + # large size like by avoiding source grid requirement in desc.compute. + # edit: this can be reduced once the single valued thing is fixed. @staticmethod - def compute_theta(eq, L, M=512, N=16, clebsch=None, **kwargs): + def compute_theta(eq, M=512, N=16, rho=1.0, clebsch=None, **kwargs): """Return DESC coordinates θ of Fourier Chebyshev basis nodes. The Fourier spectrum of θ in α is wider than the Chebyshev spectrum @@ -444,18 +448,18 @@ def compute_theta(eq, L, M=512, N=16, clebsch=None, **kwargs): ---------- eq : Equilibrium Equilibrium to use defining the coordinate mapping. - L : int or jnp.ndarray - Number of flux surfaces uniformly in [0, 1] on which to compute. - May also be an array of non-uniform coordinates. M : int Grid resolution in poloidal direction for Clebsch coordinate grid. Preferably power of 2. N : int Grid resolution in toroidal direction for Clebsch coordinate grid. Preferably power of 2. + rho : float or jnp.ndarray + Flux surfaces labels in [0, 1] on which to compute. clebsch : jnp.ndarray Optional, Clebsch coordinate tensor-product grid (ρ, α, ζ). - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + ``FourierChebyshevSeries.nodes(M,N,L,domain=(0,2*jnp.pi))``. + If given, ``rho`` is ignored. kwargs Additional parameters to supply to the coordinate mapping function. See ``desc.equilibrium.Equilibrium.map_coordinates``. @@ -465,14 +469,14 @@ def compute_theta(eq, L, M=512, N=16, clebsch=None, **kwargs): theta : jnp.ndarray Shape (L, M, N). DESC coordinates θ sourced from the Clebsch coordinates - ``FourierChebyshevBasis.nodes(M,N,L,domain=(0,2*jnp.pi))``. + ``FourierChebyshevSeries.nodes(M,N,L,domain=(0,2*jnp.pi))``. """ if clebsch is None: clebsch = FourierChebyshevSeries.nodes( check_posint(M), check_posint(N), - L, + rho, domain=(0, 2 * jnp.pi), ) return eq.map_coordinates( @@ -528,9 +532,9 @@ def points(self, pitch_inv, *, num_well=None): ------- z1, z2 : tuple[jnp.ndarray] Shape (L, num_pitch, num_well). - ζ coordinates of bounce points. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in the - epigraph of |B|. + Tuple of length two (z1, z2) that stores ζ coordinates of bounce points. + The points are ordered and grouped such that the straight line path + between ``z1`` and ``z2`` resides in the epigraph of |B|. If there were less than ``num_well`` wells detected along a field line, then the last axis, which enumerates bounce points for a particular field @@ -549,9 +553,10 @@ def check_points(self, points, pitch_inv, *, plot=True, **kwargs): ---------- points : tuple[jnp.ndarray] Shape (L, num_pitch, num_well). - ζ coordinates of bounce points. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in the - epigraph of |B|. + Output of method ``self.points``. + Tuple of length two (z1, z2) that stores ζ coordinates of bounce points. + The points are ordered and grouped such that the straight line path + between ``z1`` and ``z2`` resides in the epigraph of |B|. pitch_inv : jnp.ndarray Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by @@ -569,6 +574,7 @@ def check_points(self, points, pitch_inv, *, plot=True, **kwargs): Matplotlib (fig, ax) tuples for the 1D plot of each field line. """ + kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") return self._B.check_intersect1d( z1=_swap_pl(points[0]), z2=_swap_pl(points[1]), @@ -580,8 +586,8 @@ def check_points(self, points, pitch_inv, *, plot=True, **kwargs): def integrate( self, integrand, - points, pitch_inv, + points=None, f=None, weight=None, *, @@ -600,16 +606,17 @@ def integrate( accept the arrays in ``f`` as arguments as well as the additional keyword arguments: ``B`` and ``pitch``. A quadrature will be performed to approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. - points : tuple[jnp.ndarray] - Shape (L, num_pitch, num_well). - ζ coordinates of bounce points. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in the - epigraph of |B|. pitch_inv : jnp.ndarray Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by ``pitch_inv[ρ]`` where in the latter the labels are interpreted as the indices that correspond to that field line. + points : tuple[jnp.ndarray] + Shape (L, num_pitch, num_well). + Output of method ``self.points``. + Tuple of length two (z1, z2) that stores ζ coordinates of bounce points. + The points are ordered and grouped such that the straight line path + between ``z1`` and ``z2`` resides in the epigraph of |B|. f : list[jnp.ndarray] or jnp.ndarray Shape (L, m, n). Real scalar-valued (2π × 2π) periodic in (θ, ζ) functions evaluated @@ -676,6 +683,10 @@ def _integrate(self, integrand, points, pitch_inv, f, check, plot): axes=(-1, -2), ) B = self._B.eval1d(zeta) + # TODO: not all f is single-valued. effective ripple, Gamma_c is, Gamma_c + # valasco is not. FFT single valued part than sum afterword with + # multivalued correction part longer term solution may be adding to + # compute metadata single valued term and multivalued correction factor f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] result = _swap_pl( (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) @@ -733,6 +744,9 @@ def plot(self, l, pitch_inv=None, **kwargs): kwargs["z1"] = z1 kwargs["z2"] = z2 kwargs["k"] = pitch_inv + kwargs.setdefault( + "hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$" + ) fig, ax = B.plot1d(B.cheb, **_set_default_plot_kwargs(kwargs)) return fig, ax diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 2bfabb355..81791a2b8 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -532,6 +532,8 @@ def _check_interp(shape, Q, b_sup_z, B, result, f, plot): # Number of those integrals that were computed. actual = (marked & jnp.isfinite(result)).sum() + # fine for this fail as long as it's low, but check again after + # multivaluedness thing fixed assert goal == actual, ( f"Lost {goal - actual} integrals from NaN generation in the integrand. This " "is caused by floating point error, usually due to a poor quadrature choice." diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 9fb487c50..ffc20ec58 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -66,10 +66,9 @@ def cheb_pts(N, domain=(-1, 1), lobatto=False): def fourier_pts(M): - """Get ``M`` Fourier points.""" - m = jnp.arange(1, M + 1) - # [0, 2π] instead of [-π, π] to match our choice of α - return 2 * jnp.pi * m / M + """Get ``M`` Fourier points in [0, 2π].""" + # [0, 2π] instead of [-π, π] required to match our definition of α. + return 2 * jnp.pi * jnp.arange(M) / M def harmonic(a, M, axis=-1): diff --git a/desc/integrals/quad_utils.py b/desc/integrals/quad_utils.py index cdcc2609a..60ce1b8de 100644 --- a/desc/integrals/quad_utils.py +++ b/desc/integrals/quad_utils.py @@ -242,7 +242,7 @@ def get_quadrature(quad, automorphism): Parameters ---------- - quad : (jnp.ndarray, jnp.ndarray) + quad : tuple[jnp.ndarray] Quadrature points xₖ and weights wₖ for the approximate evaluation of the integral ∫₋₁¹ g(x) dx = ∑ₖ wₖ g(xₖ). automorphism : (Callable, Callable) or None diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 9a8403aca..c0c94c777 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1543,10 +1543,11 @@ def test_binormal_drift_bounce2d(self): # Recompute on non-symmetric, fft compatible grid. eq = things["eq"] - # FIXME: Change LinearGrid to default to Fourier points nodes. Interpolation - # fails on LinearGrid. + # FIXME: Change LinearGrid to default to Fourier points nodes. FFT + # interpolation looks gross on LinearGrid. edit: likely because + # of multi-valued ness issue, but check after done grid = Grid.create_meshgrid( - [data["rho"], fourier_pts(4 * eq._M_grid), fourier_pts(1) / eq.NFP], + [data["rho"], fourier_pts(5 * eq._M_grid), fourier_pts(1) / eq.NFP], NFP=eq.NFP, ) grid_data = eq.compute( @@ -1556,7 +1557,7 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 512, 16 + M, N = 1024, 32 # I have concern that numpy computes Nyquist frequency component incorrectly, # and the reason high Fourier resolution is required is that with large sample # frequency the incorrect max frequency component is higher than the max @@ -1566,14 +1567,16 @@ def test_binormal_drift_bounce2d(self): data=grid_data, theta=Bounce2D.compute_theta( eq, + M, + N, data["rho"], - M=M, - N=N, iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), - N_B=2 * N, - num_transit=4, - alpha=data["alpha"], # - 2 * np.pi * data["iota"], + N_B=N, + num_transit=5, + # need to include commented shift but first make sure multivaluedness + # is fixed in interpolation by checking plot of f_0. + alpha=data["alpha"], # - 2 * np.pi * data["iota"] - 0.1, Bref=data["Bref"], Lref=data["a"], check=True, @@ -1585,29 +1588,32 @@ def test_binormal_drift_bounce2d(self): f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) drift_numerical_num = bounce.integrate( integrand=TestBounce1D.drift_num_integrand, - points=points, pitch_inv=pitch_inv, + points=points, f=f, check=True, plot=True, ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, - points=points, pitch_inv=pitch_inv, + points=points, check=True, plot=True, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg + + # FIXME: Bug found and confirmed. due to fft of multivalued function gbdrift # # np.testing.assert_allclose( # noqa: E800 # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 # ) # noqa: E800 fig, ax = plt.subplots() - ax.plot(pitch_inv, drift_analytic) - ax.plot(pitch_inv, drift_numerical) + ax.plot(pitch_inv, drift_analytic, label="analytic") + ax.plot(pitch_inv, drift_numerical, label="numerical") + plt.legend() plt.show() return fig From 38e71e9d55b26a3df17c06291a9398a9454e575f Mon Sep 17 00:00:00 2001 From: unalmis Date: Tue, 24 Sep 2024 21:14:13 -0400 Subject: [PATCH 20/44] Marking test xfail --- desc/integrals/basis.py | 16 ++++++++-------- desc/integrals/interp_utils.py | 2 +- tests/test_integrals.py | 11 +++++++---- 3 files changed, 16 insertions(+), 13 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index eb47860e6..5e4972627 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -172,7 +172,7 @@ def nodes(M, N, L=None, domain=(-1, 1), lobatto=False): Optional, resolution in radial direction of domain [0, 1]. May also be an array of coordinates values. If given, then the returned ``coords`` is a 3D tensor-product with shape (L * M * N, 3). - domain : (float, float) + domain : tuple[float] Domain for y coordinates. Default is [-1, 1]. lobatto : bool Whether to use the Gauss-Lobatto (Extrema-plus-Endpoint) @@ -477,11 +477,11 @@ def intersect1d(self, k=0.0, *, num_intersect=None, pad_value=0.0): Returns ------- - z1, z2 : (jnp.ndarray, jnp.ndarray) + z1, z2 : tuple[jnp.ndarray] Shape broadcasts with (..., *self.cheb.shape[:-2], num_intersect). - Coordinates of intersects. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in - the epigraph of f. + Tuple of length two (z1, z2) of coordinates of intersects. + The points are ordered and grouped such that the straight line path + between ``z1`` and ``z2`` resides in the epigraph of f. """ errorif( @@ -544,9 +544,9 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): ---------- z1, z2 : jnp.ndarray Shape must broadcast with (*self.cheb.shape[:-2], W). - Coordinates of intersects. The points are ordered and grouped such - that the straight line path between ``z1`` and ``z2`` resides in - the epigraph of f. + Tuple of length two (z1, z2) of coordinates of intersects. + The points are ordered and grouped such that the straight line path + between ``z1`` and ``z2`` resides in the epigraph of f. k : jnp.ndarray Shape must broadcast with self.cheb.shape[:-2]. k such that fₓ(yᵢ) = k. diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index ffc20ec58..0ff837542 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -44,7 +44,7 @@ def cheb_pts(N, domain=(-1, 1), lobatto=False): ---------- N : int Number of points. - domain : (float, float) + domain : tuple[float] Domain for points. lobatto : bool Whether to return the Gauss-Lobatto (extrema-plus-endpoint) diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 7300a8971..641b36c01 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1531,8 +1531,11 @@ def test_alpha_sequence(self, alpha_0, iota, num_period, period): assert alphas.shape == (iota.size, num_period) print(alphas) + @pytest.mark.xfail( + reason="More DESC infrastructure required to interpolate multivalued integrand." + ) @pytest.mark.unit - @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) + # @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) def test_binormal_drift_bounce2d(self): """Test bounce-averaged drift with analytical expressions.""" data, things = TestBounce1D.get_drift_analytic_data() @@ -1555,11 +1558,12 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 1024, 32 + M, N = 512, 32 # I have concern that numpy computes Nyquist frequency component incorrectly, # and the reason high Fourier resolution is required is that with large sample # frequency the incorrect max frequency component is higher than the max - # frequency of theta, hiding the mistake. + # frequency of theta, hiding the mistake. Edit: maybe it's worth pursuing + # https://github.com/jax-ml/jax/issues/23895. bounce = Bounce2D( grid=grid, data=grid_data, @@ -1604,7 +1608,6 @@ def test_binormal_drift_bounce2d(self): assert drift_numerical.size == drift_analytic.size, msg # FIXME: Bug found and confirmed. due to fft of multivalued function gbdrift - # # np.testing.assert_allclose( # noqa: E800 # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 # ) # noqa: E800 From 05cb19518651ca9fea70c7e7d912aad4a17b5607 Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 25 Sep 2024 03:02:02 -0400 Subject: [PATCH 21/44] Adding more tests and documentation --- desc/integrals/basis.py | 6 ++- desc/integrals/bounce_integral.py | 78 +++++++++++++++++++++++++++--- desc/integrals/bounce_utils.py | 11 +++-- tests/test_integrals.py | 79 +++++++++++++++++++++++++++++-- 4 files changed, 158 insertions(+), 16 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 5e4972627..b2a247790 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -255,7 +255,10 @@ def compute_cheb(self, x): """ # Add axis to broadcast against Chebyshev coefficients. x = jnp.atleast_1d(x)[..., jnp.newaxis] - cheb = cheb_from_dct(irfft_non_uniform(x, self._c, self.M, axis=-2)) + # Add axis to broadcast against multiple x values. + cheb = cheb_from_dct( + irfft_non_uniform(x, self._c[..., jnp.newaxis, :, :], self.M, axis=-2) + ) assert cheb.shape[-2:] == (x.shape[-2], self.N) return PiecewiseChebyshevSeries(cheb, self.domain) @@ -510,7 +513,6 @@ def intersect1d(self, k=0.0, *, num_intersect=None, pad_value=0.0): # ``_eps`` from ``domain[-1]``. Edit: For differentiability, we cannot # consider intersects at boundary of Chebyshev polynomial. Again, cases # where this would be incorrect have measure zero. - # TODO: (kaya) look into that plot with M=512, N=32 is_z1 = (df_dy_sign <= 0) & is_intersect is_z2 = (df_dy_sign >= 0) & _in_epigraph_and(is_intersect, df_dy_sign) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index d4f72d255..2b559b716 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -290,6 +290,7 @@ class Bounce2D(IOAble): -------- Bounce1D Uses one-dimensional local spline methods for the same task. + Does not assume G is single-valued in θ and ζ. Below are some advantages of ``Bounce2D`` over ``Bounce1D``. The coordinates on which the root-finding must be done to map from DESC @@ -638,7 +639,7 @@ def integrate( Returns ------- result : jnp.ndarray - Shape is same as input points. + Shape (M, L, num_pitch, num_well). Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. @@ -717,12 +718,12 @@ def plot(self, l, pitch_inv=None, **kwargs): ---------- l : int Index into the nodes of the grid supplied to make this object. - ``rho=grid.meshgrid_reshape(grid.nodes[:,0],"rtz")[l,0,0]``. + ``rho=grid.compress(grid.nodes[:,0])[l]``. pitch_inv : jnp.ndarray Shape (num_pitch, ). Optional, 1/λ values whose corresponding bounce points on the field line specified by Clebsch coordinate ρ(l) will be plotted. - kwargs : dict + kwargs Keyword arguments into ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. @@ -744,12 +745,75 @@ def plot(self, l, pitch_inv=None, **kwargs): kwargs["z1"] = z1 kwargs["z2"] = z2 kwargs["k"] = pitch_inv - kwargs.setdefault( - "hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$" - ) + kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") fig, ax = B.plot1d(B.cheb, **_set_default_plot_kwargs(kwargs)) return fig, ax + def plot_theta(self, l, **kwargs): + """Plot θ(α, ζ) on the specified flux surface. + + Notes + ----- + The definition of α in B = ∇ρ × ∇α on an irrational magnetic surface + implies the true map θ(α, ζ) is multivalued. In particular, following an + irrational field, θ grows to ∞ (always non-monotonically) as ζ → ∞. + Therefore, it is impossible to approximate this map using single-valued + basis functions defined on a bounded subset of ℝ². + + However, our interest lies in computing functions which are + periodic in θ, so it suffices to interpolate θ over one branch cut. + DESC chooses the branch cut defined by (α, ζ) ∈ [0, 2π]² and we must + maintain that convention with our basis functions. On such a branch + cut, the bound θ ∈ [0, 4π] holds. + + Likewise, α is multivalued. As the field line is followed, the label + jumps to α ∉ [0, 2π] after completing some toroidal transit. Therefore, + the map θ(α, ζ) must be periodic in α with period 2π. At every point + ζₚ ∈ [2π k, 2π ℓ] where k, ℓ ∈ ℤ where the field line completes a + poloidal transit there is guaranteed to exist a discrete jump + discontinuity in θ at ζ = 2π ℓ(p), starting the toroidal transit. + Recall a jump discontinuity appears as an infinitely sharp cut; + nearby the cut, the function must be blind to the cut. + + Parameters + ---------- + l : int + Index into the nodes of the grid supplied to make this object. + ``rho=grid.compress(grid.nodes[:,0])[l]``. + kwargs + Keyword arguments into + ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. + + Returns + ------- + fig, ax + Matplotlib (fig, ax) tuple. + + """ + T = self._T + if T.cheb.ndim > 2: + T = PiecewiseChebyshevSeries(T.cheb[l], T.domain) + kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") + fig, ax = T.plot1d( + T.cheb, **_set_default_plot_kwargs(kwargs, vlabel=r"$\theta$") + ) + return fig, ax + # Note: + # It is simple to reconstruct the true map θ(α, ζ) from our + # function that approximates over one the branch cut. At every + # ζ = 2π ℓ, ℓ ∈ ℤ add either 0 or 2π or 4π to the next cut + # toroidal cut, where this number is chosen to ensure continuity. + # Can evaluate θ left and right of ζ = 2π ℓ to determine the amount + # to add. This is only useful if one would like to interpolate + # functions that are multi-valued in θ, ζ such as gbdrift. + # still for such functions one needs to use a 1D non-periodic + # interpolation such as in ``Bounce1D`` or split the function + # into a single-valued part, and a multivalued part where the + # multivalued part can be evaluated accurately anywhere given + # some object that can evaluate the single valued part anywhere. + # The infrastructure required for this would be to add such + # information to data_index. + class Bounce1D(IOAble): """Computes bounce integrals using one-dimensional local spline methods. @@ -1088,7 +1152,7 @@ def integrate( Returns ------- result : jnp.ndarray - Shape is same as input points. + Shape (M, L, num_pitch, num_well). Last axis enumerates the bounce integrals for a given field line, flux surface, and pitch value. diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index e2b906004..25e20267d 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -234,15 +234,18 @@ def bounce_points( return z1, z2 -def _set_default_plot_kwargs(kwargs): +def _set_default_plot_kwargs(kwargs, vlabel=r"$\vert B \vert$"): kwargs.setdefault( "title", - r"Intersects $\zeta$ in epigraph($\vert B \vert$) s.t. " - r"$\vert B \vert(\zeta) = 1/\lambda$", + r"Intersects $\zeta$ in epigraph(" + + vlabel + + ") s.t. " + + vlabel + + r"$(\zeta) = 1/\lambda$", ) kwargs.setdefault("klabel", r"$1/\lambda$") kwargs.setdefault("hlabel", r"$\zeta$") - kwargs.setdefault("vlabel", r"$\vert B \vert$") + kwargs.setdefault("vlabel", vlabel) return kwargs diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 641b36c01..f06795714 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1177,8 +1177,6 @@ def test_bounce1d_checks(self): # 9. Example manipulation of the output # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) - # Group the result by pitch and flux surface. - result = result.reshape(alpha.size, rho.size, pitch_inv.shape[-1]) # The result stored at m, l, p = 0, 1, 3 print("Result(α, ρ, λ):", result[m, l, p]) @@ -1186,7 +1184,7 @@ def test_bounce1d_checks(self): print("1/λ(α, ρ):", pitch_inv[l, p]) # for the Clebsch-type field line coordinates nodes = grid.source_grid.meshgrid_reshape(grid.source_grid.nodes[:, :2], "arz") - print("(α, ρ):", nodes[m, l, 0]) + print("(ρ, α):", nodes[m, l, 0]) # 10. Plotting fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=False) @@ -1531,6 +1529,81 @@ def test_alpha_sequence(self, alpha_0, iota, num_period, period): assert alphas.shape == (iota.size, num_period) print(alphas) + @pytest.mark.unit + @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d * 4) + def test_bounce2d_checks(self): + """Test that all the internal correctness checks pass for real example.""" + # noqa: D202 + # Suppose we want to compute a bounce average of the function + # f(ℓ) = (1 − λ|B|/2) * g_zz, where g_zz is the squared norm of the + # toroidal basis vector on some set of field lines specified by (ρ, α) + # coordinates. This is defined as + # [∫ f(ℓ) / √(1 − λ|B|) dℓ] / [∫ 1 / √(1 − λ|B|) dℓ] + + # 1. Define python functions for the integrands. We do that above. + # 2. Pick flux surfaces and grid resolution. + rho = np.linspace(0.1, 1, 6) + eq = get("HELIOTRON") + grid = Grid.create_meshgrid( + [rho, fourier_pts(3 * eq._M_grid), fourier_pts(3 * eq.N_grid) / eq.NFP], + NFP=eq.NFP, + ) + # 3. Compute input data. + data = eq.compute( + Bounce2D.required_names + ["min_tz |B|", "max_tz |B|", "g_zz"], grid=grid + ) + + # 4. Compute DESC coordinates of optimal interpolation nodes. + theta = Bounce2D.compute_theta(eq, M=256, N=32, rho=rho) + + # 5. Make the bounce integration operator. + bounce = Bounce2D( + grid, data, theta, num_transit=2, N_B=128, quad=leggauss(3), check=True + ) + pitch_inv, _ = bounce.get_pitch_inv_quad( + min_B=grid.compress(data["min_tz |B|"]), + max_B=grid.compress(data["max_tz |B|"]), + num_pitch=10, + ) + # 6. Compute bounce points. + points = bounce.points(pitch_inv) + # 7. Optionally check for correctness of bounce points. + bounce.check_points(points, pitch_inv, plot=False) + # 8. Integrate. + num = bounce.integrate( + integrand=TestBounce1D._example_numerator, + pitch_inv=pitch_inv, + f=Bounce2D.reshape_data(grid, data["g_zz"]), + points=points, + check=True, + ) + den = bounce.integrate( + integrand=TestBounce1D._example_denominator, + pitch_inv=pitch_inv, + points=points, + check=True, + ) + avg = safediv(num, den) + assert np.isfinite(avg).all() and np.count_nonzero(avg) + + # 9. Example manipulation of the output + # Sum all bounce averages across a particular field line, for every field line. + result = avg.sum(axis=-1) + # The result stored at + l, p = 1, 3 + print("Result(ρ, λ):", result[l, p]) + # corresponds to the 1/λ value + print("1/λ(ρ):", pitch_inv[l, p]) + # for the flux surface + print("ρ:", rho[l]) + + # 10. Plotting + fig, ax = bounce.plot_theta(l) + # TODO: why does this converge to different |B| than test_bounce1d_checks? + fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) + plt.show() + return fig + @pytest.mark.xfail( reason="More DESC infrastructure required to interpolate multivalued integrand." ) From 134d82662d690b96e839019152ecd4ee8d21fd65 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 26 Sep 2024 18:56:42 -0400 Subject: [PATCH 22/44] short-circuit convergence of poloidal Fourier series --- desc/integrals/basis.py | 70 ++++++++++++-------------- desc/integrals/bounce_integral.py | 84 ++++++++++++++----------------- desc/integrals/interp_utils.py | 11 ++-- tests/test_integrals.py | 15 ++++-- 4 files changed, 88 insertions(+), 92 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index b2a247790..d180df651 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -112,35 +112,29 @@ class FourierChebyshevSeries(IOAble): Performance may improve significantly if the spectral resolutions ``M`` and ``N`` are powers of two. + + Parameters + ---------- + f : jnp.ndarray + Shape (..., M, N). + Samples of real function on the ``FourierChebyshevSeries.nodes`` grid. + domain : tuple[float] + Domain for y coordinates. Default is [-1, 1]. + lobatto : bool + Whether ``f`` was sampled on the Gauss-Lobatto (extrema-plus-endpoint) + instead of the interior roots grid for Chebyshev points. + Attributes ---------- M : int Fourier spectral resolution. N : int Chebyshev spectral resolution. - domain : (float, float) - Domain for y coordinates. - lobatto : bool - Whether ``f`` was sampled on the Gauss-Lobatto (extrema-plus-endpoint) - instead of the interior roots grid for Chebyshev points. """ def __init__(self, f, domain=(-1, 1), lobatto=False): - """Interpolate Fourier-Chebyshev series to ``f``. - - Parameters - ---------- - f : jnp.ndarray - Shape (..., M, N). - Samples of real function on the ``FourierChebyshevSeries.nodes`` grid. - domain : (float, float) - Domain for y coordinates. Default is [-1, 1]. - lobatto : bool - Whether ``f`` was sampled on the Gauss-Lobatto (extrema-plus-endpoint) - instead of the interior roots grid for Chebyshev points. - - """ + """Interpolate Fourier-Chebyshev series to ``f``.""" self.M = f.shape[-2] self.N = f.shape[-1] errorif(domain[0] > domain[-1], msg="Got inverted domain.") @@ -269,32 +263,18 @@ class PiecewiseChebyshevSeries(IOAble): { fₓ | fₓ : y ↦ ∑ₙ₌₀ᴺ⁻¹ aₙ(x) Tₙ(y) } and Tₙ are Chebyshev polynomials on [−yₘᵢₙ, yₘₐₓ] - Attributes + Parameters ---------- cheb : jnp.ndarray Shape (..., M, N). - Chebyshev coefficients αₙ(x) for fₓ(y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). - M : int - Number of cuts. - N : int - Chebyshev spectral resolution of each cut. - domain : (float, float) - Domain for y coordinates. + Chebyshev coefficients αₙ(x) for f(x, y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). + domain : tuple[float] + Domain for y coordinates. Default is [-1, 1]. """ def __init__(self, cheb, domain=(-1, 1)): - """Make piecewise series from given Chebyshev coefficients. - - Parameters - ---------- - cheb : jnp.ndarray - Shape (..., M, N). - Chebyshev coefficients αₙ(x=``x``) for f(x, y) = ∑ₙ₌₀ᴺ⁻¹ αₙ(x) Tₙ(y). - domain : (float, float) - Domain for y coordinates. Default is [-1, 1]. - - """ + """Make piecewise series from given Chebyshev coefficients.""" self.cheb = jnp.atleast_2d(cheb) errorif(domain[0] > domain[-1], msg="Got inverted domain.") self.domain = tuple(domain) @@ -309,6 +289,20 @@ def N(self): """Chebyshev spectral resolution.""" return self.cheb.shape[-1] + def stitch(self): + """Enforce the piecewise series is continuous.""" + # This is used to short-circuit convergence of poloidal Fourier series + # so that |B| along the field line is perfectly continuous. + # Useful at lower resolution because the singularities in + # the bounce integrals are strong functions of |B| alone. + + # evaluate at left boundary + f_0 = self.cheb[..., ::2].sum(axis=-1) - self.cheb[..., 1::2].sum(axis=-1) + # evaluate at right boundary + f_1 = self.cheb.sum(axis=-1) + dfx = f_1[..., :-1] - f_0[..., 1:] # Δf = f(xᵢ, y₁) - f(xᵢ₊₁, y₀) + self.cheb = self.cheb.at[..., 1:, 0].add(dfx) # + f(xᵢ₊₁, y₀) + def evaluate(self, N): """Evaluate Chebyshev series at N Chebyshev points. diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 2b559b716..564239e39 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -1,5 +1,7 @@ """Methods for computing bounce integrals (singular or otherwise).""" +from copy import deepcopy + import numpy as np from interpax import CubicHermiteSpline, PPoly from numpy.fft import rfft2 @@ -153,6 +155,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): ).reshape(*alpha.shape, N_B) # Need |B| parameterized by single variable to compute roots. B = PiecewiseChebyshevSeries(cheb_from_dct(dct(B, type=2, axis=-1)) / N_B, T.domain) + B.stitch() return T, B @@ -190,7 +193,7 @@ class Bounce2D(IOAble): Brief description of algorithm. Magnetic field line with label α, defined by B = ∇ρ × ∇α, is determined from - α : ρ, θ, ζ ↦ θ + λ(ρ,θ,ζ) − ι(ρ) [ζ + ω(ρ,θ,ζ)] + α : ρ, θ, ζ ↦ θ + λ(ρ,θ,ζ) − ι(ρ) [ζ + ω(ρ,θ,ζ)] Interpolate Fourier-Chebyshev series to DESC poloidal coordinate. θ : α, ζ ↦ tₘₙ exp(jmα) Tₙ(ζ) Compute |B| along field lines. @@ -212,12 +215,11 @@ class Bounce2D(IOAble): which is invariant to the order of summation). The DESC coordinate system is related to field-line-following coordinate - systems by a relation whose solution is best found with Newton iteration. - There is a unique real solution to that relation, so Newton iteration is a - globally convergent root-finding algorithm here. For the task of finding - bounce points, Newton iteration is not a globally convergent algorithm to - find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following - coordinate. For this, function approximation of |B| is necessary. + systems by a relation whose solution is best found with Newton iteration + since this solution is unique. Newton iteration is not a globally + convergent algorithm to find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where + ζ is a field-line-following coordinate. For this, function approximation + of |B| is necessary. Therefore, to compute bounce points {(ζ₁, ζ₂)}, we approximate |B| by a series expansion of basis functions parameterized by a single variable ζ, @@ -290,7 +292,6 @@ class Bounce2D(IOAble): -------- Bounce1D Uses one-dimensional local spline methods for the same task. - Does not assume G is single-valued in θ and ζ. Below are some advantages of ``Bounce2D`` over ``Bounce1D``. The coordinates on which the root-finding must be done to map from DESC @@ -322,7 +323,8 @@ def __init__( N_B=32, num_transit=32, # TODO: Allow multiple starting labels for near-rational surfaces. - # think can just concatenate along second to last axis of cheb + # think can just concatenate along second to last axis of cheb. + # Do this in different PR. alpha=0.0, quad=leggauss(32), automorphism=(automorphism_sin, grad_automorphism_sin), @@ -605,8 +607,9 @@ def integrate( The composition operator on the set of functions in ``f`` that maps the functions in ``f`` to the integrand f(λ, ℓ) in ∫ f(λ, ℓ) dℓ. It should accept the arrays in ``f`` as arguments as well as the additional keyword - arguments: ``B`` and ``pitch``. A quadrature will be performed to - approximate the bounce integral of ``integrand(*f,B=B,pitch=pitch)``. + arguments: ``B``, ``pitch``, and ``zeta``. A quadrature will be performed + to approximate the bounce integral of + ``integrand(*f,B=B,pitch=pitch,zeta=zeta)``. pitch_inv : jnp.ndarray Shape (L, num_pitch). 1/λ values to compute the bounce integrals. 1/λ(ρ) is specified by @@ -684,25 +687,21 @@ def _integrate(self, integrand, points, pitch_inv, f, check, plot): axes=(-1, -2), ) B = self._B.eval1d(zeta) - # TODO: not all f is single-valued. effective ripple, Gamma_c is, Gamma_c - # valasco is not. FFT single valued part than sum afterword with - # multivalued correction part longer term solution may be adding to - # compute metadata single valued term and multivalued correction factor f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] result = _swap_pl( - (integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis]) / b_sup_z) + ( + integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis], zeta=zeta) + / b_sup_z + ) .reshape(shape) .dot(self._w) * grad_bijection_from_disc(z1, z2) ) if check: - num_pitch = shape[0] - num_rho = shape[-3] - shape[0] = num_rho - shape[-3] = num_pitch + shape[-3], shape[0] = shape[0], shape[-3] _check_interp( - # num_alpha, num_rho, num_pitch, num_well, num_quad + # num_alpha is 1, num_rho, num_pitch, num_well, num_quad (1, *shape), *map(_swap_pl, (zeta, b_sup_z, B)), result, @@ -749,16 +748,18 @@ def plot(self, l, pitch_inv=None, **kwargs): fig, ax = B.plot1d(B.cheb, **_set_default_plot_kwargs(kwargs)) return fig, ax - def plot_theta(self, l, **kwargs): + def plot_theta(self, l, stitch=False, **kwargs): """Plot θ(α, ζ) on the specified flux surface. Notes ----- The definition of α in B = ∇ρ × ∇α on an irrational magnetic surface - implies the true map θ(α, ζ) is multivalued. In particular, following an - irrational field, θ grows to ∞ (always non-monotonically) as ζ → ∞. - Therefore, it is impossible to approximate this map using single-valued - basis functions defined on a bounded subset of ℝ². + implies the angle θ(α, ζ) is multivalued. In particular, following an + irrational field, the single-valued θ grows to ∞ + (always non-monotonically) as ζ → ∞. Therefore, it is impossible to + approximate this map using single-valued basis functions defined on a + bounded subset of ℝ² (recall continuous functions on compact sets attain + their maximum). However, our interest lies in computing functions which are periodic in θ, so it suffices to interpolate θ over one branch cut. @@ -780,6 +781,8 @@ def plot_theta(self, l, **kwargs): l : int Index into the nodes of the grid supplied to make this object. ``rho=grid.compress(grid.nodes[:,0])[l]``. + stitch : bool + Whether to plot the single-valued θ(α, ζ). kwargs Keyword arguments into ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. @@ -791,6 +794,9 @@ def plot_theta(self, l, **kwargs): """ T = self._T + if stitch: + T = deepcopy(T) + T.stitch() if T.cheb.ndim > 2: T = PiecewiseChebyshevSeries(T.cheb[l], T.domain) kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") @@ -798,21 +804,6 @@ def plot_theta(self, l, **kwargs): T.cheb, **_set_default_plot_kwargs(kwargs, vlabel=r"$\theta$") ) return fig, ax - # Note: - # It is simple to reconstruct the true map θ(α, ζ) from our - # function that approximates over one the branch cut. At every - # ζ = 2π ℓ, ℓ ∈ ℤ add either 0 or 2π or 4π to the next cut - # toroidal cut, where this number is chosen to ensure continuity. - # Can evaluate θ left and right of ζ = 2π ℓ to determine the amount - # to add. This is only useful if one would like to interpolate - # functions that are multi-valued in θ, ζ such as gbdrift. - # still for such functions one needs to use a 1D non-periodic - # interpolation such as in ``Bounce1D`` or split the function - # into a single-valued part, and a multivalued part where the - # multivalued part can be evaluated accurately anywhere given - # some object that can evaluate the single valued part anywhere. - # The infrastructure required for this would be to add such - # information to data_index. class Bounce1D(IOAble): @@ -843,12 +834,11 @@ class Bounce1D(IOAble): which is invariant to the order of summation). The DESC coordinate system is related to field-line-following coordinate - systems by a relation whose solution is best found with Newton iteration. - There is a unique real solution to that relation, so Newton iteration is a - globally convergent root-finding algorithm here. For the task of finding - bounce points, Newton iteration is not a globally convergent algorithm to - find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where ζ is a field-line-following - coordinate. For this, function approximation of |B| is necessary. + systems by a relation whose solution is best found with Newton iteration + since this solution is unique. Newton iteration is not a globally + convergent algorithm to find the real roots of r : ζ ↦ |B|(ζ) − 1/λ where + ζ is a field-line-following coordinate. For this, function approximation + of |B| is necessary. The function approximation in ``Bounce1D`` is ignorant that the objects to approximate are defined on a bounded subset of ℝ². Instead, the domain is diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 0ff837542..80233d783 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -11,7 +11,7 @@ import numpy as np from interpax import interp1d -from orthax.chebyshev import chebroots, chebvander +from orthax.chebyshev import chebroots from desc.backend import dct, jnp, rfft, rfft2, take from desc.integrals.quad_utils import bijection_from_disc @@ -393,9 +393,12 @@ def idct_non_uniform(xq, a, n, axis=-1): """ a = jnp.moveaxis(a, axis, -1) - # Could use Clenshaw recursion with fq = chebval(xq, a, tensor=False). - basis = chebvander(xq, n - 1) - fq = jnp.linalg.vecdot(basis, a) + # Equivalent to + # Clenshaw recursion: chebval(xq, a, tensor=False), + # Vandermode product: jnp.linalg.vecdot(chebvander(xq, n - 1), a) + # but performs better on GPU. + n = jnp.arange(n) + fq = jnp.linalg.vecdot(jnp.cos(n * jnp.arccos(xq)[..., jnp.newaxis]), a) return fq diff --git a/tests/test_integrals.py b/tests/test_integrals.py index f06795714..cda298825 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1529,6 +1529,15 @@ def test_alpha_sequence(self, alpha_0, iota, num_period, period): assert alphas.shape == (iota.size, num_period) print(alphas) + @staticmethod + def _example_numerator(g_zz, B, pitch, zeta): + f = (1 - 0.5 * pitch * B) * g_zz + return safediv(f, jnp.sqrt(jnp.abs(1 - pitch * B))) + + @staticmethod + def _example_denominator(B, pitch, zeta): + return safediv(1, jnp.sqrt(jnp.abs(1 - pitch * B))) + @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d * 4) def test_bounce2d_checks(self): @@ -1571,14 +1580,14 @@ def test_bounce2d_checks(self): bounce.check_points(points, pitch_inv, plot=False) # 8. Integrate. num = bounce.integrate( - integrand=TestBounce1D._example_numerator, + integrand=TestBounce2D._example_numerator, pitch_inv=pitch_inv, f=Bounce2D.reshape_data(grid, data["g_zz"]), points=points, check=True, ) den = bounce.integrate( - integrand=TestBounce1D._example_denominator, + integrand=TestBounce2D._example_denominator, pitch_inv=pitch_inv, points=points, check=True, @@ -1598,7 +1607,7 @@ def test_bounce2d_checks(self): print("ρ:", rho[l]) # 10. Plotting - fig, ax = bounce.plot_theta(l) + fig, ax = bounce.plot_theta(l, stitch=True) # TODO: why does this converge to different |B| than test_bounce1d_checks? fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) plt.show() From 455353fa94d400e0e63a291a65ce78907e62a38c Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 26 Sep 2024 19:05:24 -0400 Subject: [PATCH 23/44] Remove resolved comment --- desc/integrals/bounce_utils.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 25e20267d..80059891c 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -536,8 +536,6 @@ def _check_interp(shape, Q, b_sup_z, B, result, f, plot): # Number of those integrals that were computed. actual = (marked & jnp.isfinite(result)).sum() - # fine for this fail as long as it's low, but check again after - # multivaluedness thing fixed assert goal == actual, ( f"Lost {goal - actual} integrals from NaN generation in the integrand. This " "is caused by floating point error, usually due to a poor quadrature choice." From f63f3b3d73405fe5e4998d60ad3d8faf13080962 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 26 Sep 2024 19:33:23 -0400 Subject: [PATCH 24/44] Stitch theta as well for robustness --- desc/integrals/bounce_integral.py | 75 ++++++++++++++++--------------- tests/test_integrals.py | 2 +- 2 files changed, 41 insertions(+), 36 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 564239e39..eba27d12d 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -1,7 +1,5 @@ """Methods for computing bounce integrals (singular or otherwise).""" -from copy import deepcopy - import numpy as np from interpax import CubicHermiteSpline, PPoly from numpy.fft import rfft2 @@ -144,6 +142,11 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): # Evaluating set of single variable maps is more efficient than evaluating # multivariable map, so we project θ to a set of Chebyshev series. T = FourierChebyshevSeries(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) + # Theoretically, stitching θ is only necessary to recover single-valued ness + # necessary to bounce integrate quantities that are multivalued at a physical + # location. However, it is still useful to do so to ensure continuity is not + # lost when projecting the 2D basis to 1D. + T.stitch() theta = T.evaluate(N_B) xq = jnp.stack( [theta, jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape)], axis=-1 @@ -205,6 +208,11 @@ class Bounce2D(IOAble): Perform Gaussian quadrature after removing singularities. Fᵢ : λ, ζ₁, ζ₂ ↦ ∫ᵢ f(λ, ζ, {Gⱼ}) dζ + If the map G is multivalued at a physical location, then it is still + permissible if separable into a single valued and multivalued parts. + In that case, supply the single valued parts, which will be interpolated + with FFTs, and use the provided coordinates θ,ζ ∈ ℝ to compose G. + Longer description for developers. For applications which reduce to computing a nonlinear function of distance @@ -288,6 +296,33 @@ class Bounce2D(IOAble): quadrature packs nodes at reasonable density. Fast multipoint methods are preferable because they are exact, but that requires more development work. + Additional notes on multivalued coordinates. + The definition of α in B = ∇ρ × ∇α on an irrational magnetic surface + implies the angle θ(α, ζ) is multivalued at a physical location. + In particular, following an irrational field, the single-valued θ grows + to ∞ (always non-monotonically) as ζ → ∞. Therefore, it is impossible to + approximate this map using single-valued basis functions defined on a + bounded subset of ℝ² (recall continuous functions on compact sets attain + their maximum). + + Still, it suffices to interpolate θ over one branch cut. + DESC chooses the branch cut defined by (α, ζ) ∈ [0, 2π]² and we must + maintain that convention with our basis functions. On such a branch + cut, the bound θ ∈ [0, 4π] holds. + + Likewise, α is multivalued. As the field line is followed, the label + jumps to α ∉ [0, 2π] after completing some toroidal transit. Therefore, + the map θ(α, ζ) must be periodic in α with period 2π. At every point + ζₚ ∈ [2π k, 2π ℓ] where k, ℓ ∈ ℤ where the field line completes a + poloidal transit there is guaranteed to exist a discrete jump + discontinuity in θ at ζ = 2π ℓ(p), starting the toroidal transit. + Recall a jump discontinuity appears as an infinitely sharp cut; + nearby the cut, the function must be blind to the cut. + + To recover the single-valued θ(α, ζ) from the function approximation + over one branch cut, at every ζ = 2π ℓ we can add either 0 or 2π or + 4π to the next cut of θ. + See Also -------- Bounce1D @@ -300,8 +335,8 @@ class Bounce2D(IOAble): ``Bounce1D`` requires ``L*M*N*num_transit``. Furthermore, pseudo-spectral interpolation of smooth functions such as |B| on each flux surface is more efficient. This reduces the number of bounce integrals to be done by an - order of magnitude. Furthermore, C1 cubic spline interpolation is - inefficient to reconstruct smooth local maxima of |B|, which might be + order of magnitude. Also, we have noticed C1 cubic spline interpolation + is inefficient to reconstruct smooth local maxima of |B|, which might be important for (strongly) singular bounce integrals whose estimation depends on ∂|B|/∂ζ there. @@ -748,41 +783,14 @@ def plot(self, l, pitch_inv=None, **kwargs): fig, ax = B.plot1d(B.cheb, **_set_default_plot_kwargs(kwargs)) return fig, ax - def plot_theta(self, l, stitch=False, **kwargs): + def plot_theta(self, l, **kwargs): """Plot θ(α, ζ) on the specified flux surface. - Notes - ----- - The definition of α in B = ∇ρ × ∇α on an irrational magnetic surface - implies the angle θ(α, ζ) is multivalued. In particular, following an - irrational field, the single-valued θ grows to ∞ - (always non-monotonically) as ζ → ∞. Therefore, it is impossible to - approximate this map using single-valued basis functions defined on a - bounded subset of ℝ² (recall continuous functions on compact sets attain - their maximum). - - However, our interest lies in computing functions which are - periodic in θ, so it suffices to interpolate θ over one branch cut. - DESC chooses the branch cut defined by (α, ζ) ∈ [0, 2π]² and we must - maintain that convention with our basis functions. On such a branch - cut, the bound θ ∈ [0, 4π] holds. - - Likewise, α is multivalued. As the field line is followed, the label - jumps to α ∉ [0, 2π] after completing some toroidal transit. Therefore, - the map θ(α, ζ) must be periodic in α with period 2π. At every point - ζₚ ∈ [2π k, 2π ℓ] where k, ℓ ∈ ℤ where the field line completes a - poloidal transit there is guaranteed to exist a discrete jump - discontinuity in θ at ζ = 2π ℓ(p), starting the toroidal transit. - Recall a jump discontinuity appears as an infinitely sharp cut; - nearby the cut, the function must be blind to the cut. - Parameters ---------- l : int Index into the nodes of the grid supplied to make this object. ``rho=grid.compress(grid.nodes[:,0])[l]``. - stitch : bool - Whether to plot the single-valued θ(α, ζ). kwargs Keyword arguments into ``desc/integrals/basis.py::PiecewiseChebyshevSeries.plot1d``. @@ -794,9 +802,6 @@ def plot_theta(self, l, stitch=False, **kwargs): """ T = self._T - if stitch: - T = deepcopy(T) - T.stitch() if T.cheb.ndim > 2: T = PiecewiseChebyshevSeries(T.cheb[l], T.domain) kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") diff --git a/tests/test_integrals.py b/tests/test_integrals.py index cda298825..526b24223 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1607,7 +1607,7 @@ def test_bounce2d_checks(self): print("ρ:", rho[l]) # 10. Plotting - fig, ax = bounce.plot_theta(l, stitch=True) + fig, ax = bounce.plot_theta(l) # TODO: why does this converge to different |B| than test_bounce1d_checks? fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) plt.show() From b1de0d635ac74ccaf41b7ecb5d207dfe13b095e5 Mon Sep 17 00:00:00 2001 From: unalmis Date: Fri, 27 Sep 2024 02:07:09 -0400 Subject: [PATCH 25/44] Solved the convergence issue --- desc/integrals/basis.py | 7 +----- desc/integrals/bounce_integral.py | 41 +++++++++++++++++-------------- tests/test_integrals.py | 37 +++++++++++++++------------- 3 files changed, 43 insertions(+), 42 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index d180df651..a36df24fc 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -291,17 +291,12 @@ def N(self): def stitch(self): """Enforce the piecewise series is continuous.""" - # This is used to short-circuit convergence of poloidal Fourier series - # so that |B| along the field line is perfectly continuous. - # Useful at lower resolution because the singularities in - # the bounce integrals are strong functions of |B| alone. - # evaluate at left boundary f_0 = self.cheb[..., ::2].sum(axis=-1) - self.cheb[..., 1::2].sum(axis=-1) # evaluate at right boundary f_1 = self.cheb.sum(axis=-1) dfx = f_1[..., :-1] - f_0[..., 1:] # Δf = f(xᵢ, y₁) - f(xᵢ₊₁, y₀) - self.cheb = self.cheb.at[..., 1:, 0].add(dfx) # + f(xᵢ₊₁, y₀) + self.cheb = self.cheb.at[..., 1:, 0].add(dfx.cumsum(axis=-1)) def evaluate(self, N): """Evaluate Chebyshev series at N Chebyshev points. diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index eba27d12d..140063299 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -110,6 +110,24 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): """Transform to single variable Clebsch spectral domain. + Notes + ----- + It appears the Fourier transform of θ may have small oscillatory bumps outside + reasonable bandwidths. This impedes full convergence of θ(α, ζ=0). Maybe this + is because θ is not a physical signal as it is determined by the optimizer. + + The field line label changes discontinuously, so functions defined on + cuts of the field line are not guaranteed continuity until full convergence. + Therefore, we explicitly enforce continuity of θ to short-circuit the + convergence. + + This works well because the first cut is on α=0, which is a knot of the + Fourier series, the Chebyshev points include a knot near endpoints, + so θ at the next cut of the field line is known with precision. + (Moreso if using Lobatto nodes). Note the Fourier series converges fast + for |B|, even in non-omnigenous configurations where (∂|B|/∂α)|ρ,ζ is not + small, so this is indeed some feature with θ. + Parameters ---------- grid : Grid @@ -142,10 +160,6 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): # Evaluating set of single variable maps is more efficient than evaluating # multivariable map, so we project θ to a set of Chebyshev series. T = FourierChebyshevSeries(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) - # Theoretically, stitching θ is only necessary to recover single-valued ness - # necessary to bounce integrate quantities that are multivalued at a physical - # location. However, it is still useful to do so to ensure continuity is not - # lost when projecting the 2D basis to 1D. T.stitch() theta = T.evaluate(N_B) xq = jnp.stack( @@ -156,9 +170,9 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): f=B[..., jnp.newaxis, :, :], axes=(-1, -2), ).reshape(*alpha.shape, N_B) - # Need |B| parameterized by single variable to compute roots. + # Parameterize |B| by single variable to compute roots. B = PiecewiseChebyshevSeries(cheb_from_dct(dct(B, type=2, axis=-1)) / N_B, T.domain) - B.stitch() + # |B| guaranteed to be continuous because it was interpolated from B(θ(α, ζ),ζ). return T, B @@ -356,7 +370,7 @@ def __init__( data, theta, N_B=32, - num_transit=32, + num_transit=16, # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb. # Do this in different PR. @@ -467,21 +481,10 @@ def __init__( assert self._T.N == theta.shape[-1] assert self._B.N == N_B - # The Fourier series converges fast for |B|(α, ζ); - # for near omnigenous configurations in particular, (∂|B|/∂α)|ρ,ζ vanishes. - # However, the Fourier series of θ(α, ζ) converges slower than desired. - # Small discontinuities of quantities evaluated between adjacent cuts of a - # field line vanish at M > 256. Memory seems much better than DESC's transforms - # though, and we avoid creation of a 1000 intermediate dependency arrays of - # large size like by avoiding source grid requirement in desc.compute. - # edit: this can be reduced once the single valued thing is fixed. @staticmethod - def compute_theta(eq, M=512, N=16, rho=1.0, clebsch=None, **kwargs): + def compute_theta(eq, M=32, N=16, rho=1.0, clebsch=None, **kwargs): """Return DESC coordinates θ of Fourier Chebyshev basis nodes. - The Fourier spectrum of θ in α is wider than the Chebyshev spectrum - of θ in ζ, so M > N is recommended. - Parameters ---------- eq : Equilibrium diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 526b24223..97e275bfd 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1563,11 +1563,11 @@ def test_bounce2d_checks(self): ) # 4. Compute DESC coordinates of optimal interpolation nodes. - theta = Bounce2D.compute_theta(eq, M=256, N=32, rho=rho) + theta = Bounce2D.compute_theta(eq, M=64, N=32, rho=rho) # 5. Make the bounce integration operator. bounce = Bounce2D( - grid, data, theta, num_transit=2, N_B=128, quad=leggauss(3), check=True + grid, data, theta, num_transit=3, N_B=64, quad=leggauss(3), check=True ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), @@ -1613,9 +1613,17 @@ def test_bounce2d_checks(self): plt.show() return fig - @pytest.mark.xfail( - reason="More DESC infrastructure required to interpolate multivalued integrand." - ) + @staticmethod + def drift_num_integrand(cvdrift, gbdrift, B, pitch, zeta): + """Integrand of numerator of bounce averaged binormal drift.""" + g = jnp.sqrt(1 - pitch * B) + return (cvdrift * g) - (0.5 * g * gbdrift) + (0.5 * gbdrift / g) + + @staticmethod + def drift_den_integrand(B, pitch, zeta): + """Integrand of denominator of bounce averaged binormal drift.""" + return 1 / jnp.sqrt(1 - pitch * B) + @pytest.mark.unit # @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) def test_binormal_drift_bounce2d(self): @@ -1640,12 +1648,7 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 512, 32 - # I have concern that numpy computes Nyquist frequency component incorrectly, - # and the reason high Fourier resolution is required is that with large sample - # frequency the incorrect max frequency component is higher than the max - # frequency of theta, hiding the mistake. Edit: maybe it's worth pursuing - # https://github.com/jax-ml/jax/issues/23895. + M, N = 16, 16 bounce = Bounce2D( grid=grid, data=grid_data, @@ -1657,10 +1660,10 @@ def test_binormal_drift_bounce2d(self): iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), N_B=N, - num_transit=5, - # need to include commented shift but first make sure multivaluedness - # is fixed in interpolation by checking plot of f_0. - alpha=data["alpha"], # - 2 * np.pi * data["iota"] - 0.1, + num_transit=3, + # can check if multivalued quantity interpolated correctly + # by comparing plot of f_0 with bounce1d. + alpha=data["alpha"] - 2 * np.pi * data["iota"], Bref=data["Bref"], Lref=data["a"], check=True, @@ -1671,7 +1674,7 @@ def test_binormal_drift_bounce2d(self): f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) drift_numerical_num = bounce.integrate( - integrand=TestBounce1D.drift_num_integrand, + integrand=TestBounce2D.drift_num_integrand, pitch_inv=pitch_inv, f=f, points=points, @@ -1679,7 +1682,7 @@ def test_binormal_drift_bounce2d(self): plot=True, ) drift_numerical_den = bounce.integrate( - integrand=TestBounce1D.drift_den_integrand, + integrand=TestBounce2D.drift_den_integrand, pitch_inv=pitch_inv, points=points, check=True, From e028106c68ab8121421111814c2ff39d2b90c127 Mon Sep 17 00:00:00 2001 From: unalmis Date: Fri, 27 Sep 2024 03:02:46 -0400 Subject: [PATCH 26/44] support vector valued to simplify multivalued function reconstruction --- desc/integrals/bounce_integral.py | 29 ++++++++++++++++++++++++----- tests/test_integrals.py | 12 +++++++----- 2 files changed, 31 insertions(+), 10 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 140063299..965120175 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -1,6 +1,5 @@ """Methods for computing bounce integrals (singular or otherwise).""" -import numpy as np from interpax import CubicHermiteSpline, PPoly from numpy.fft import rfft2 from orthax.legendre import leggauss @@ -629,6 +628,7 @@ def integrate( integrand, pitch_inv, f=None, + f_vec=None, weight=None, points=None, *, @@ -659,6 +659,12 @@ def integrate( on the ``grid`` supplied to construct this object. These functions should be arguments to the callable ``integrand``. Use the method ``Bounce2D.reshape_data`` to reshape the data into the expected shape. + f_vec : list[jnp.ndarray] or jnp.ndarray + Shape (L, m, n, 3). + Real vector-valued (2π × 2π) periodic in (θ, ζ) functions evaluated + on the ``grid`` supplied to construct this object. These functions + should be arguments to the callable ``integrand``. Use the method + ``Bounce2D.reshape_data`` to reshape the data into the expected shape. weight : jnp.ndarray Shape (L, m, n). If supplied, the bounce integral labeled by well j is weighted such that @@ -687,15 +693,18 @@ def integrate( """ errorif(weight is not None, NotImplementedError, msg="See Bounce1D") f = setdefault(f, []) + f_vec = setdefault(f_vec, []) if not isinstance(f, (list, tuple)): - f = [f] if isinstance(f, (jnp.ndarray, np.ndarray)) else list(f) + f = [f] + if not isinstance(f_vec, (list, tuple)): + f_vec = [f_vec] points = map(_swap_pl, points) pitch_inv = atleast_nd(self._B.cheb.ndim - 1, pitch_inv).T - result = self._integrate(integrand, points, pitch_inv, f, check, plot) + result = self._integrate(integrand, points, pitch_inv, f, f_vec, check, plot) return result - def _integrate(self, integrand, points, pitch_inv, f, check, plot): + def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): """Bounce integrate ∫ f(λ, ℓ) dℓ. Parameters @@ -706,6 +715,8 @@ def _integrate(self, integrand, points, pitch_inv, f, check, plot): Shape (num_pitch, ) or (num_pitch, L). f : list[jnp.ndarray] Shape (m, n) or (L, m, n). + f : list[jnp.ndarray] + Shape (m, n, 3) or (L, m, n, 3). """ z1, z2 = points @@ -726,9 +737,17 @@ def _integrate(self, integrand, points, pitch_inv, f, check, plot): ) B = self._B.eval1d(zeta) f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] + f_vec = [ + interp_rfft2( + Q[..., jnp.newaxis, :], f_i[..., jnp.newaxis, :, :, :], axes=(-2, -3) + ) + for f_i in f_vec + ] result = _swap_pl( ( - integrand(*f, B=B, pitch=1 / pitch_inv[..., jnp.newaxis], zeta=zeta) + integrand( + *f, *f_vec, B=B, pitch=1 / pitch_inv[..., jnp.newaxis], zeta=zeta + ) / b_sup_z ) .reshape(shape) diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 97e275bfd..ab90e3850 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1673,10 +1673,12 @@ def test_binormal_drift_bounce2d(self): bounce.check_points(points, pitch_inv, plot=True) f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) + f_vec = Bounce2D.reshape_data(grid, grid_data["grad(alpha)"]) drift_numerical_num = bounce.integrate( integrand=TestBounce2D.drift_num_integrand, pitch_inv=pitch_inv, f=f, + f_vec=f_vec, points=points, check=True, plot=True, @@ -1692,14 +1694,14 @@ def test_binormal_drift_bounce2d(self): msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg - # FIXME: Bug found and confirmed. due to fft of multivalued function gbdrift - # np.testing.assert_allclose( # noqa: E800 - # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 - # ) # noqa: E800 - fig, ax = plt.subplots() ax.plot(pitch_inv, drift_analytic, label="analytic") ax.plot(pitch_inv, drift_numerical, label="numerical") plt.legend() plt.show() + + # np.testing.assert_allclose( # noqa: E800 + # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 + # ) # noqa: E800 + return fig From 2ce57ee1c2b99dcacee0f3b6659c11f3f386a6ff Mon Sep 17 00:00:00 2001 From: unalmis Date: Fri, 27 Sep 2024 13:14:33 -0400 Subject: [PATCH 27/44] Fix comment --- desc/integrals/quad_utils.py | 2 +- tests/test_integrals.py | 4 ++-- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/desc/integrals/quad_utils.py b/desc/integrals/quad_utils.py index b3a3d1b6b..4f70dec61 100644 --- a/desc/integrals/quad_utils.py +++ b/desc/integrals/quad_utils.py @@ -58,7 +58,7 @@ def automorphism_sin(x, s=0, m=10): """[-1, 1] ∋ x ↦ y ∈ [−1, 1]. This map increases node density near the boundary by the asymptotic factor - 1/√(1−x²) and adds a √(1−x²) factor to the integrand. + 1/√(1−x²) and adds a cosine factor to the integrand. Parameters ---------- diff --git a/tests/test_integrals.py b/tests/test_integrals.py index ab90e3850..aa82d868c 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -959,14 +959,14 @@ def test_bounce_quadrature(self, is_strong, quad, automorphism): Notes ----- Empirical testing shows asymptotic density of nodes needs to be at least - 1/√(1−x²) and quadrature needs √(1−x²) factor in Jacobian for accurate + 1/√(1−x²) and quadrature needs a cosine factor in Jacobian for accurate bounce integrals. This is satisfied by ``chebgauss2`` and ``leggauss`` with the sin automorphism. The former has less clustering near boundary by a factor of 1/√(1−x²), so we choose it for weakly singular bounce integrals. This will capture more features in the integral, especially the W shaped wells. Less clustering will also make non-uniform FFTs more accurate. - For the strongly singular bounce integrals, another √(1−x²) factor is preferred + For the strongly singular bounce integrals, another cosine factor is preferred to supress the derivative (as expected from chain rule), so we need to use the sin automorphism. We choose to apply that map to ``leggauss`` instead of ``_chebgauss1`` because the extra cosine term in ``_chebgauss1`` increases the From 7b1267a37cf15afd3f0de086eac64b12759f85c0 Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 28 Sep 2024 21:36:37 -0400 Subject: [PATCH 28/44] Add explanation of math in c --- desc/integrals/bounce_integral.py | 56 ++++++++++++++++++++++--------- 1 file changed, 41 insertions(+), 15 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 965120175..0ae8c6ead 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -111,21 +111,47 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Notes ----- - It appears the Fourier transform of θ may have small oscillatory bumps outside - reasonable bandwidths. This impedes full convergence of θ(α, ζ=0). Maybe this - is because θ is not a physical signal as it is determined by the optimizer. - - The field line label changes discontinuously, so functions defined on - cuts of the field line are not guaranteed continuity until full convergence. - Therefore, we explicitly enforce continuity of θ to short-circuit the - convergence. - - This works well because the first cut is on α=0, which is a knot of the - Fourier series, the Chebyshev points include a knot near endpoints, - so θ at the next cut of the field line is known with precision. - (Moreso if using Lobatto nodes). Note the Fourier series converges fast - for |B|, even in non-omnigenous configurations where (∂|B|/∂α)|ρ,ζ is not - small, so this is indeed some feature with θ. + The field line label α changes discontinuously, so the approximation + g defined with basis function in (α, ζ) coordinates to some continuous + function f does not guarantee continuity between cuts of the field line + until full convergence of g to f. + + Note if g were defined with basis functions in straight field line + coordinates, then continuity between cuts of the field line, as + determined by the straight field line coordinates (ϑ, ζ), is + guaranteed even with incomplete convergence (because the + parameters (ϑ, ζ) change continuously along the field line). + + Do not interpret this as superior function approximation. + Indeed, if g is defined with basis functions in (α, ζ) coordinates, then + g(α=α₀, ζ) will sample the approximation to f(α=α₀, ζ) for the full domain in ζ. + This holds even with incomplete convergence of g to f. + However, if g is defined with basis functions in (ϑ, ζ) coordinates, then + g(ϑ(α=α₀,ζ), ζ) will sample the approximation f(α=α₀ ± ε, ζ) with ε → 0 as + g converges to f. + + (Visually, the small discontinuity apparent in g(α, ζ) at cuts of the field + line will not be visible in g(ϑ, ζ) because when moving along the field line + with g(ϑ, ζ) one is continuously flowing away from the starting field line, + (whereas g(α, ζ) has to "decide" at the cut what the next field line is). + (If full convergence is difficult to achieve, then in the context of surface + averaging bounce integrals, function approximation in (α, ζ) coordinates + might be preferable because most of the bounce integrals do not stretch + across toroidal transits).) + + Now, it appears the Fourier transform of θ may have small oscillatory bumps + outside reasonable bandwidths. This impedes full convergence of any + approximation, and in particular the poloidal Fourier series for, θ(α, ζ=ζ₀). + Maybe this is because θ is not a physical signal as it is determined by the + optimizer. (Note the Fourier series converges fast for |B|, even in + non-omnigenous configurations where (∂|B|/∂α)|ρ,ζ is not small, so this is + indeed some feature with θ). + + Therefore, we explicitly enforce continuity of our approximation of θ between + cuts to short-circuit the convergence. This works to remove the small + discontinuity between cuts of the field line because the first cut is on α=0, + which is a knot of the Fourier series, and the Chebyshev points include a knot + near endpoints, so θ at the next cut of the field line is known with precision. Parameters ---------- From 47988d1a78b0ba5abcb6713d17c6631a29fa3e22 Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 28 Sep 2024 21:40:28 -0400 Subject: [PATCH 29/44] small but important typo in comment --- desc/integrals/bounce_integral.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 0ae8c6ead..968921095 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -127,7 +127,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): g(α=α₀, ζ) will sample the approximation to f(α=α₀, ζ) for the full domain in ζ. This holds even with incomplete convergence of g to f. However, if g is defined with basis functions in (ϑ, ζ) coordinates, then - g(ϑ(α=α₀,ζ), ζ) will sample the approximation f(α=α₀ ± ε, ζ) with ε → 0 as + g(ϑ(α=α₀,ζ), ζ) will sample the approximation to f(α=α₀ ± ε, ζ) with ε → 0 as g converges to f. (Visually, the small discontinuity apparent in g(α, ζ) at cuts of the field From ea227a1015ca8cbd49b39c040c495484cc9036de Mon Sep 17 00:00:00 2001 From: unalmis Date: Tue, 1 Oct 2024 20:14:09 -0400 Subject: [PATCH 30/44] Genealize to NFP!=1 --- desc/integrals/bounce_integral.py | 43 ++++++++++++------ desc/integrals/interp_utils.py | 59 ++++++++++++++++-------- tests/test_integrals.py | 23 +++++----- tests/test_interp_utils.py | 74 ++++++++++++++++++++----------- 4 files changed, 131 insertions(+), 68 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 968921095..c02b26df8 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -48,8 +48,7 @@ def _transform_to_desc(grid, f, is_reshaped=False): Parameters ---------- grid : Grid - Tensor-product grid in (θ, ζ) with uniformly spaced nodes in - (2π × 2π) poloidal and toroidal coordinates. + Tensor-product grid in (θ, ζ) with uniformly spaced nodes [0, 2π) × [0, 2π/NFP). Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. f : jnp.ndarray Function evaluated on ``grid``. @@ -73,8 +72,7 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): Parameters ---------- grid : Grid - Tensor-product grid in (θ, ζ) with uniformly spaced nodes in - (2π × 2π) poloidal and toroidal coordinates. + Tensor-product grid in (θ, ζ) with uniformly spaced nodes [0, 2π) × [0, 2π/NFP). Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. nodes : jnp.ndarray Shape (L, M, N, 2) or (M, N, 2). @@ -100,6 +98,7 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): # which is not a tensor product node set in DESC space. xq=nodes.reshape(*nodes.shape[:-3], M * N, 2), f=f[..., jnp.newaxis, :, :], + domain1=(0, 2 * jnp.pi / grid.NFP), axes=(-1, -2), ).reshape(*nodes.shape[:-3], M, N), domain=(0, 2 * jnp.pi), @@ -156,8 +155,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Parameters ---------- grid : Grid - Tensor-product grid in (θ, ζ) with uniformly spaced nodes in - (2π × 2π) poloidal and toroidal coordinates. + Tensor-product grid in (θ, ζ) with uniformly spaced nodes [0, 2π) × [0, 2π/NFP). Preferably power of 2 for ``grid.num_theta`` and ``grid.num_zeta``. alpha : jnp.ndarray Shape (L, num_transit) or (num_transit, ). @@ -193,6 +191,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): B = interp_rfft2( xq=xq, f=B[..., jnp.newaxis, :, :], + domain1=(0, 2 * jnp.pi / grid.NFP), axes=(-1, -2), ).reshape(*alpha.shape, N_B) # Parameterize |B| by single variable to compute roots. @@ -421,9 +420,8 @@ def __init__( Parameters ---------- grid : Grid - Tensor-product grid in (ρ, θ, ζ) with uniformly spaced nodes in - (2π × 2π) poloidal and toroidal coordinates. - Note that below shape notation defines + Tensor-product grid in (ρ, θ, ζ) with uniformly spaced nodes + [0, 2π) × [0, 2π/NFP). Note that below shape notation defines L = ``grid.num_rho``, m = ``grid.num_theta``, and n = ``grid.num_zeta``. data : dict[str, jnp.ndarray] Data evaluated on ``grid``. @@ -480,6 +478,7 @@ def __init__( self._alpha = alpha self._m = grid.num_theta self._n = grid.num_zeta + self._NFP = grid.NFP self._x, self._w = get_quadrature(quad, automorphism) # peel off field lines @@ -748,10 +747,12 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): z1, z2 = points shape = [*z1.shape, self._x.size] + # This is ζ along the field line. zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) ) - # Expects shape (num_pitch, L) if T.cheb.shape[0] is L. + # Note self._T expects shape (num_pitch, L) if T.cheb.shape[0] is L. + # These are the (θ, ζ) coordinates of the quadrature points. Q = jnp.stack([self._T.eval1d(zeta), zeta], axis=-1) b_sup_z = irfft2_non_uniform( @@ -759,20 +760,36 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): a=self._b_sup_z[..., jnp.newaxis, :, :], M=self._n, N=self._m, + domain1=(0, 2 * jnp.pi / self._NFP), axes=(-1, -2), ) B = self._B.eval1d(zeta) - f = [interp_rfft2(Q, f_i[..., jnp.newaxis, :, :], axes=(-1, -2)) for f_i in f] + f = [ + interp_rfft2( + Q, + f_i[..., jnp.newaxis, :, :], + domain1=(0, 2 * jnp.pi / self._NFP), + axes=(-1, -2), + ) + for f_i in f + ] f_vec = [ interp_rfft2( - Q[..., jnp.newaxis, :], f_i[..., jnp.newaxis, :, :, :], axes=(-2, -3) + Q[..., jnp.newaxis, :], + f_i[..., jnp.newaxis, :, :, :], + domain1=(0, 2 * jnp.pi / self._NFP), + axes=(-2, -3), ) for f_i in f_vec ] result = _swap_pl( ( integrand( - *f, *f_vec, B=B, pitch=1 / pitch_inv[..., jnp.newaxis], zeta=zeta + *f, + *f_vec, + B=B, + pitch=1 / pitch_inv[..., jnp.newaxis], + zeta=zeta, ) / b_sup_z ) diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 80233d783..e2f33066f 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -66,7 +66,7 @@ def cheb_pts(N, domain=(-1, 1), lobatto=False): def fourier_pts(M): - """Get ``M`` Fourier points in [0, 2π].""" + """Get ``M`` Fourier points in [0, 2π).""" # [0, 2π] instead of [-π, π] required to match our definition of α. return 2 * jnp.pi * jnp.arange(M) / M @@ -112,7 +112,7 @@ def harmonic(a, M, axis=-1): return h -def harmonic_vander(x, M): +def harmonic_vander(x, M, domain=(0, 2 * np.pi)): """Nyquist trigonometric interpolant basis evaluated at ``x``. Parameters @@ -121,6 +121,9 @@ def harmonic_vander(x, M): Points at which to evaluate pseudo-Vandermonde matrix. M : int Spectral resolution. + domain : tuple[float] + Domain over which samples will be taken. + This domain should span an open period of the function to interpolate. Returns ------- @@ -130,8 +133,8 @@ def harmonic_vander(x, M): Last axis ordered as [1, cos(x), ..., cos(mx), sin(x), sin(2x), ..., sin(mx)]. """ - m = jnp.fft.rfftfreq(M, d=1 / M) - mx = m * x[..., jnp.newaxis] + m = jnp.fft.rfftfreq(M, d=np.diff(domain) / (2 * jnp.pi) / M) + mx = m * (x - domain[0])[..., jnp.newaxis] basis = jnp.concatenate( [jnp.cos(mx), jnp.sin(mx[..., 1 : m.size - ((M % 2) == 0)])], axis=-1 ) @@ -150,7 +153,7 @@ def harmonic_vander(x, M): # https://github.com/flatironinstitute/jax-finufft. -def interp_rfft(xq, f, axis=-1): +def interp_rfft(xq, f, domain=(0, 2 * jnp.pi), axis=-1): """Interpolate real-valued ``f`` to ``xq`` with FFT. Parameters @@ -159,7 +162,9 @@ def interp_rfft(xq, f, axis=-1): Real query points where interpolation is desired. Shape of ``xq`` must broadcast with arrays of shape ``np.delete(f.shape,axis)``. f : jnp.ndarray - Real 2π periodic function values on uniform grid to interpolate. + Real function values on uniform grid over an open period to interpolate. + domain : tuple[float] + Domain over which samples were taken. axis : int Axis along which to transform. @@ -170,12 +175,12 @@ def interp_rfft(xq, f, axis=-1): """ a = rfft(f, axis=axis, norm="forward") - fq = irfft_non_uniform(xq, a, f.shape[axis], axis) + fq = irfft_non_uniform(xq, a, f.shape[axis], domain, axis) return fq -def irfft_non_uniform(xq, a, n, axis=-1): - """Evaluate Fourier coefficients ``a`` at ``xq`` ∈ [0, 2π]. +def irfft_non_uniform(xq, a, n, domain=(0, 2 * jnp.pi), axis=-1): + """Evaluate Fourier coefficients ``a`` at ``xq``. Parameters ---------- @@ -186,6 +191,8 @@ def irfft_non_uniform(xq, a, n, axis=-1): Fourier coefficients ``a=rfft(f,axis=axis,norm="forward")``. n : int Spectral resolution of ``a``. + domain : tuple[float] + Domain over which samples were taken. axis : int Axis along which to transform. @@ -203,14 +210,17 @@ def irfft_non_uniform(xq, a, n, axis=-1): .at[..., -1] .divide(1.0 + ((n % 2) == 0)) ) - m = jnp.fft.rfftfreq(n, d=1 / n) + m = jnp.fft.rfftfreq(n, d=np.diff(domain) / (2 * jnp.pi) / n) + xq = xq - domain[0] basis = jnp.exp(-1j * m * xq[..., jnp.newaxis]) fq = 2.0 * jnp.linalg.vecdot(basis, a).real # ℜ〈 basis, a 〉= cos(m xq)⋅ℜ(a) − sin(m xq)⋅ℑ(a) return fq -def interp_rfft2(xq, f, axes=(-2, -1)): +def interp_rfft2( + xq, f, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) +): """Interpolate real-valued ``f`` to ``xq`` with FFT. Parameters @@ -224,8 +234,11 @@ def interp_rfft2(xq, f, axes=(-2, -1)): across axis ``min(axes)`` (``max(axes)``) of the function values ``f``. f : jnp.ndarray Shape (..., f.shape[-2], f.shape[-1]). - Real (2π × 2π) periodic function values on uniform tensor-product grid - to interpolate. + Real function values on uniform tensor-product grid over an open period. + domain0 : tuple[float] + Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. + domain1 : tuple[float] + Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. axes : tuple[int, int] Axes along which to transform. The real transform is done along ``axes[1]``, so it will be more @@ -238,12 +251,16 @@ def interp_rfft2(xq, f, axes=(-2, -1)): """ a = rfft2(f, axes=axes, norm="forward") - fq = irfft2_non_uniform(xq, a, f.shape[axes[0]], f.shape[axes[1]], axes) + fq = irfft2_non_uniform( + xq, a, f.shape[axes[0]], f.shape[axes[1]], domain0, domain1, axes + ) return fq -def irfft2_non_uniform(xq, a, M, N, axes=(-2, -1)): - """Evaluate Fourier coefficients ``a`` at ``xq`` ∈ [0, 2π]². +def irfft2_non_uniform( + xq, a, M, N, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) +): + """Evaluate Fourier coefficients ``a`` at ``xq``. Parameters ---------- @@ -262,6 +279,10 @@ def irfft2_non_uniform(xq, a, M, N, axes=(-2, -1)): Spectral resolution of ``a`` along ``axes[0]``. N : int Spectral resolution of ``a`` along ``axes[1]``. + domain0 : tuple[float] + Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. + domain1 : tuple[float] + Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. axes : tuple[int, int] Axes along which to transform. @@ -283,9 +304,11 @@ def irfft2_non_uniform(xq, a, M, N, axes=(-2, -1)): .divide(1.0 + ((N % 2) == 0)) ) - m = jnp.fft.fftfreq(M, d=1 / M) - n = jnp.fft.rfftfreq(N, d=1 / N) + m = jnp.fft.fftfreq(M, d=np.diff(domain0) / (2 * jnp.pi) / M) + n = jnp.fft.rfftfreq(N, d=np.diff(domain1) / (2 * jnp.pi) / N) idx = np.argsort(axes) + + xq = xq - jnp.array([domain0[0], domain1[0]]) basis = jnp.exp( 1j * ( diff --git a/tests/test_integrals.py b/tests/test_integrals.py index aa82d868c..ba8c7cefb 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1187,7 +1187,7 @@ def test_bounce1d_checks(self): print("(ρ, α):", nodes[m, l, 0]) # 10. Plotting - fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=False) + fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=True) return fig @pytest.mark.unit @@ -1554,7 +1554,7 @@ def test_bounce2d_checks(self): rho = np.linspace(0.1, 1, 6) eq = get("HELIOTRON") grid = Grid.create_meshgrid( - [rho, fourier_pts(3 * eq._M_grid), fourier_pts(3 * eq.N_grid) / eq.NFP], + [rho, fourier_pts(eq._M_grid), fourier_pts(eq.N_grid) / eq.NFP], NFP=eq.NFP, ) # 3. Compute input data. @@ -1563,11 +1563,11 @@ def test_bounce2d_checks(self): ) # 4. Compute DESC coordinates of optimal interpolation nodes. - theta = Bounce2D.compute_theta(eq, M=64, N=32, rho=rho) + theta = Bounce2D.compute_theta(eq, M=128, N=200, rho=rho) # 5. Make the bounce integration operator. bounce = Bounce2D( - grid, data, theta, num_transit=3, N_B=64, quad=leggauss(3), check=True + grid, data, theta, num_transit=1, N_B=400, quad=leggauss(3), check=True ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), @@ -1577,7 +1577,7 @@ def test_bounce2d_checks(self): # 6. Compute bounce points. points = bounce.points(pitch_inv) # 7. Optionally check for correctness of bounce points. - bounce.check_points(points, pitch_inv, plot=False) + bounce.check_points(points, pitch_inv, plot=True) # 8. Integrate. num = bounce.integrate( integrand=TestBounce2D._example_numerator, @@ -1608,8 +1608,9 @@ def test_bounce2d_checks(self): # 10. Plotting fig, ax = bounce.plot_theta(l) - # TODO: why does this converge to different |B| than test_bounce1d_checks? - fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) + fig, ax = bounce.plot( + l, pitch_inv[l], include_legend=False, show=True, num=10000 + ) plt.show() return fig @@ -1638,7 +1639,11 @@ def test_binormal_drift_bounce2d(self): # interpolation looks gross on LinearGrid. edit: likely because # of multi-valued ness issue, but check after done grid = Grid.create_meshgrid( - [data["rho"], fourier_pts(5 * eq._M_grid), fourier_pts(1) / eq.NFP], + [ + data["rho"], + fourier_pts(eq._M_grid), + fourier_pts(max(1, eq.N_grid)) / eq.NFP, + ], NFP=eq.NFP, ) grid_data = eq.compute( @@ -1673,12 +1678,10 @@ def test_binormal_drift_bounce2d(self): bounce.check_points(points, pitch_inv, plot=True) f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) - f_vec = Bounce2D.reshape_data(grid, grid_data["grad(alpha)"]) drift_numerical_num = bounce.integrate( integrand=TestBounce2D.drift_num_integrand, pitch_inv=pitch_inv, f=f, - f_vec=f_vec, points=points, check=True, plot=True, diff --git a/tests/test_interp_utils.py b/tests/test_interp_utils.py index fddc9f5d0..10f8be26c 100644 --- a/tests/test_interp_utils.py +++ b/tests/test_interp_utils.py @@ -150,8 +150,10 @@ def _f_2d(x, y): def _f_2d_nyquist_freq(): - x_freq_nyquist = 3 + 2 - y_freq_nyquist = 5 + 3 + # can just sum frequencies multiplied above thanks to fourier + x_freq, y_freq = 3, 5 + x_freq_nyquist = x_freq + 2 + y_freq_nyquist = y_freq + 3 return x_freq_nyquist, y_freq_nyquist @@ -189,51 +191,69 @@ def test_rfftfreq(self, M): @pytest.mark.unit @pytest.mark.parametrize( - "func, n", + "func, n, domain", [ - (_f_1d, 2 * _f_1d_nyquist_freq() + 1), - (_f_1d, 2 * _f_1d_nyquist_freq()), + (_f_1d, 2 * _f_1d_nyquist_freq() + 1, (0, 2 * np.pi)), + # At Nyquist: + # Should pass for 1D example above, and useful since + # others don't test if nyquist frequency component is + # computed correctly since it would just be zero. + (_f_1d, 2 * _f_1d_nyquist_freq(), (0, 2 * np.pi)), + # shifted domain + (_f_1d, 2 * _f_1d_nyquist_freq() + 1, (-np.pi, np.pi)), + (_f_1d, 2 * _f_1d_nyquist_freq(), (-np.pi, np.pi)), + # faster period, can do at Nyquist rate if off phase + (lambda x: np.cos(7 * x), 1 * 2, (-np.pi / 7, np.pi / 7)), + # faster period, need above Nyquist rate if on phase + (lambda x: np.sin(7 * x), 1 * 2 + 1, (-np.pi / 7, np.pi / 7)), ], ) - def test_interp_rfft(self, func, n): + def test_interp_rfft(self, func, n, domain): """Test non-uniform FFT interpolation.""" + x = np.linspace(domain[0], domain[1], n, endpoint=False) + f = func(x) xq = np.array([7.34, 1.10134, 2.28]) - x = np.linspace(0, 2 * np.pi, n, endpoint=False) - assert not np.any(np.isclose(xq[..., np.newaxis], x)) - f, fq = func(x), func(xq) - np.testing.assert_allclose(interp_rfft(xq, f), fq) + fq = func(xq) + np.testing.assert_allclose(interp_rfft(xq, f, domain), fq) M = f.shape[-1] - np.testing.assert_allclose( - np.sum( - harmonic_vander(xq, M) * harmonic(rfft(f, norm="forward"), M), axis=-1 - ), - fq, - ) + basis = harmonic_vander(xq, M, domain) + coef = harmonic(rfft(f, norm="forward"), M) + np.testing.assert_allclose((basis * coef).sum(axis=-1), fq) - @pytest.mark.xfail(reason="Does numpy, jax, and scipy need to fix a bug with FFT?") @pytest.mark.unit @pytest.mark.parametrize( - "func, m, n", + "func, m, n, domain0, domain1", [ - (_f_2d, 2 * _f_2d_nyquist_freq()[0] + 1, 2 * _f_2d_nyquist_freq()[1] + 1), - (_f_2d, 2 * _f_2d_nyquist_freq()[0], 2 * _f_2d_nyquist_freq()[1]), + ( + _f_2d, + 2 * _f_2d_nyquist_freq()[0] + 1, + 2 * _f_2d_nyquist_freq()[1] + 1, + (0, 2 * np.pi), + (0, 2 * np.pi), + ), + ( + _f_2d, + 2 * _f_2d_nyquist_freq()[0] + 1, + 2 * _f_2d_nyquist_freq()[1] + 1, + (-np.pi / 3, 5 * np.pi / 3), + (np.pi, 3 * np.pi), + ), ], ) - def test_interp_rfft2(self, func, m, n): + def test_interp_rfft2(self, func, m, n, domain0, domain1): """Test non-uniform FFT interpolation.""" xq = np.array([[7.34, 1.10134, 2.28], [1.1, 3.78432, 8.542]]).T - x = np.linspace(0, 2 * np.pi, m, endpoint=False) - y = np.linspace(0, 2 * np.pi, n, endpoint=False) - assert not np.any(np.isclose(xq[..., 0, np.newaxis], x)) - assert not np.any(np.isclose(xq[..., 1, np.newaxis], y)) + x = np.linspace(domain0[0], domain0[1], m, endpoint=False) + y = np.linspace(domain1[0], domain1[1], n, endpoint=False) x, y = map(np.ravel, list(np.meshgrid(x, y, indexing="ij"))) truth = func(xq[..., 0], xq[..., 1]) + f = func(x, y).reshape(m, n) np.testing.assert_allclose( - interp_rfft2(xq, func(x, y).reshape(m, n), axes=(-2, -1)), + interp_rfft2(xq, f, domain0, domain1, axes=(-2, -1)), truth, ) np.testing.assert_allclose( - interp_rfft2(xq, func(x, y).reshape(m, n), axes=(-1, -2)), + interp_rfft2(xq, f, domain0, domain1, axes=(-1, -2)), truth, ) From 89762c7052c53974d14c7ec7acffbb3cc215d96c Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 2 Oct 2024 00:23:57 -0400 Subject: [PATCH 31/44] Add NFP to get_rtz_grid --- desc/equilibrium/coords.py | 1 + desc/integrals/interp_utils.py | 4 ++-- 2 files changed, 3 insertions(+), 2 deletions(-) diff --git a/desc/equilibrium/coords.py b/desc/equilibrium/coords.py index 7e50a71a6..6b2770257 100644 --- a/desc/equilibrium/coords.py +++ b/desc/equilibrium/coords.py @@ -740,6 +740,7 @@ def get_rtz_grid( desc_grid = Grid( nodes=rtz_nodes, coordinates="rtz", + NFP=eq.NFP, source_grid=grid, sort=False, jitable=jitable, diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index e2f33066f..3c48f3d65 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -239,7 +239,7 @@ def interp_rfft2( Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. domain1 : tuple[float] Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. - axes : tuple[int, int] + axes : tuple[int] Axes along which to transform. The real transform is done along ``axes[1]``, so it will be more efficient for that to denote the larger size axis in ``axes``. @@ -283,7 +283,7 @@ def irfft2_non_uniform( Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. domain1 : tuple[float] Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. - axes : tuple[int, int] + axes : tuple[int] Axes along which to transform. Returns From c3e465b320aa90536e95384d7b42e36ea29f9330 Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 2 Oct 2024 12:57:10 -0400 Subject: [PATCH 32/44] More NFP fixes --- desc/equilibrium/coords.py | 2 +- desc/grid.py | 8 ++++- desc/integrals/bounce_integral.py | 7 ++-- desc/transform.py | 37 ++++++++++++--------- tests/test_integrals.py | 54 +++++++++++++++++++++---------- tests/test_interp_utils.py | 12 ++----- 6 files changed, 75 insertions(+), 45 deletions(-) diff --git a/desc/equilibrium/coords.py b/desc/equilibrium/coords.py index 6b2770257..ebf78e622 100644 --- a/desc/equilibrium/coords.py +++ b/desc/equilibrium/coords.py @@ -740,7 +740,7 @@ def get_rtz_grid( desc_grid = Grid( nodes=rtz_nodes, coordinates="rtz", - NFP=eq.NFP, + NFP=1, source_grid=grid, sort=False, jitable=jitable, diff --git a/desc/grid.py b/desc/grid.py index 6afe3ba8d..4b7bb281e 100644 --- a/desc/grid.py +++ b/desc/grid.py @@ -671,6 +671,7 @@ class Grid(_Grid): Use np.inf to denote no periodicity. NFP : int Number of field periods (Default = 1). + Change this only if your nodes are placed within one field period. source_grid : Grid Grid from which coordinates were mapped from. sort : bool @@ -794,7 +795,8 @@ def create_meshgrid( NFP : int Number of field periods (Default = 1). Only makes sense to change from 1 if last coordinate is periodic - with some constant divided by ``NFP``. + with some constant divided by ``NFP`` and the nodes are placed + within one field period. Returns ------- @@ -916,6 +918,8 @@ class LinearGrid(_Grid): Toroidal grid resolution. NFP : int Number of field periods (Default = 1). + Change this only if your nodes are placed within one field period + or should be interpreted as spanning one field period. sym : bool True for stellarator symmetry, False otherwise (Default = False). axis : bool @@ -1011,6 +1015,8 @@ def _create_nodes( # noqa: C901 Toroidal grid resolution. NFP : int Number of field periods (Default = 1). + Only change this if your nodes are placed within one field period + or should be interpreted as spanning one field period. axis : bool True to include a point at rho=0 (default), False for rho[0] = rho[1]/2. endpoint : bool diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index c02b26df8..6b8e76987 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -188,6 +188,7 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): xq = jnp.stack( [theta, jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape)], axis=-1 ).reshape(*alpha.shape[:-1], alpha.shape[-1] * N_B, 2) + B = interp_rfft2( xq=xq, f=B[..., jnp.newaxis, :, :], @@ -680,13 +681,13 @@ def integrate( as the indices that correspond to that field line. f : list[jnp.ndarray] or jnp.ndarray Shape (L, m, n). - Real scalar-valued (2π × 2π) periodic in (θ, ζ) functions evaluated + Real scalar-valued (2π × 2π/NFP) periodic in (θ, ζ) functions evaluated on the ``grid`` supplied to construct this object. These functions should be arguments to the callable ``integrand``. Use the method ``Bounce2D.reshape_data`` to reshape the data into the expected shape. f_vec : list[jnp.ndarray] or jnp.ndarray Shape (L, m, n, 3). - Real vector-valued (2π × 2π) periodic in (θ, ζ) functions evaluated + Real vector-valued (2π × 2π/NFP) periodic in (θ, ζ) functions evaluated on the ``grid`` supplied to construct this object. These functions should be arguments to the callable ``integrand``. Use the method ``Bounce2D.reshape_data`` to reshape the data into the expected shape. @@ -764,6 +765,8 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): axes=(-1, -2), ) B = self._B.eval1d(zeta) + # Note not all functions are periodic in ζ with period 2π/NFP. + # Users will likely want GitHub issue #1287. f = [ interp_rfft2( Q, diff --git a/desc/transform.py b/desc/transform.py index a5c1798ad..e6cd6f0f3 100644 --- a/desc/transform.py +++ b/desc/transform.py @@ -8,7 +8,13 @@ from desc.backend import jnp, put from desc.io import IOAble -from desc.utils import combination_permutation, isalmostequal, islinspaced, issorted +from desc.utils import ( + combination_permutation, + errorif, + isalmostequal, + islinspaced, + issorted, +) class Transform(IOAble): @@ -59,21 +65,22 @@ def __init__( self._basis = basis self._rcond = rcond if rcond is not None else "auto" - if ( - method != "jitable" - and grid.node_pattern != "custom" + errorif( + self.grid.coordinates != "rtz", + msg=f"Expected coordinates rtz got {self.grid.coordinates}.", + ) + # DESC truncates the computational domain to ζ ∈ [0, 2π/grid.NFP] + # and changes variables to the spectrally condensed ζ* = basis.NFP ζ, + # so basis.NFP must equal grid.NFP. + errorif( + self.grid.NFP != self.basis.NFP and self.basis.N != 0 - and self.grid.NFP != self.basis.NFP - and np.any(self.grid.nodes[:, 2] != 0) - ): - warnings.warn( - colored( - "Unequal number of field periods for grid {} and basis {}.".format( - self.grid.NFP, self.basis.NFP - ), - "yellow", - ) - ) + and grid.node_pattern != "custom" + # I'd argue these two are not worth it + and method != "jitable" and np.any(self.grid.nodes[:, 2] != 0), + msg=f"Unequal number of field periods for grid {self.grid.NFP} and " + f"basis {self.basis.NFP}.", + ) self._built = False self._built_pinv = False diff --git a/tests/test_integrals.py b/tests/test_integrals.py index ba8c7cefb..dd8b8ca9b 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -57,7 +57,7 @@ from desc.integrals.singularities import _get_quadrature_nodes from desc.integrals.surface_integral import _get_grid_surface from desc.transform import Transform -from desc.utils import dot, safediv +from desc.utils import dot, errorif, safediv class TestSurfaceIntegral: @@ -1134,11 +1134,11 @@ def test_bounce1d_checks(self): # 1. Define python functions for the integrands. We do that above. # 2. Pick flux surfaces, field lines, and how far to follow the field # line in Clebsch coordinates ρ, α, ζ. - rho = np.linspace(0.1, 1, 6) - alpha = np.array([0, 0.5]) - zeta = np.linspace(-2 * np.pi, 2 * np.pi, 200) + rho = 1 # np.linspace(0.1, 1, 6) + alpha = 0 # np.array([0, 0.5]) + zeta = np.linspace(0, 2 * np.pi, 400) - eq = get("HELIOTRON") + eq = get("W7-X") # 3. Convert above coordinates to DESC computational coordinates. grid = get_rtz_grid(eq, rho, alpha, zeta, coordinates="raz") # 4. Compute input data. @@ -1146,7 +1146,9 @@ def test_bounce1d_checks(self): Bounce1D.required_names + ["min_tz |B|", "max_tz |B|", "g_zz"], grid=grid ) # 5. Make the bounce integration operator. - bounce = Bounce1D(grid.source_grid, data, quad=leggauss(3), check=True) + bounce = Bounce1D( + grid.source_grid, data, quad=leggauss(3), check=True, warn=False + ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), max_B=grid.compress(data["max_tz |B|"]), @@ -1172,13 +1174,19 @@ def test_bounce1d_checks(self): batch=False, ) avg = safediv(num, den) - assert np.isfinite(avg).all() and np.count_nonzero(avg) + errorif(not np.isfinite(avg).all()) + errorif( + np.count_nonzero(avg) == 0, + msg="Detected 0 wells on this cut of the field line. Make sure enough " + "toroidal transits were followed for this test, or plot the field line " + "to see if this is expected.", + ) # 9. Example manipulation of the output # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) # The result stored at - m, l, p = 0, 1, 3 + m, l, p = 0, 0, 3 print("Result(α, ρ, λ):", result[m, l, p]) # corresponds to the 1/λ value print("1/λ(α, ρ):", pitch_inv[l, p]) @@ -1551,10 +1559,15 @@ def test_bounce2d_checks(self): # 1. Define python functions for the integrands. We do that above. # 2. Pick flux surfaces and grid resolution. - rho = np.linspace(0.1, 1, 6) - eq = get("HELIOTRON") + rho = 1 # np.linspace(0.1, 1, 6) + eq = get("W7-X") grid = Grid.create_meshgrid( - [rho, fourier_pts(eq._M_grid), fourier_pts(eq.N_grid) / eq.NFP], + [ + rho, + fourier_pts(eq._M_grid), + fourier_pts(max(1, eq.N_grid)) / eq.NFP, + ], + period=(np.inf, 2 * np.pi, 2 * np.pi / eq.NFP), NFP=eq.NFP, ) # 3. Compute input data. @@ -1563,11 +1576,18 @@ def test_bounce2d_checks(self): ) # 4. Compute DESC coordinates of optimal interpolation nodes. - theta = Bounce2D.compute_theta(eq, M=128, N=200, rho=rho) + theta = Bounce2D.compute_theta(eq, M=2, N=512, rho=rho) # 5. Make the bounce integration operator. bounce = Bounce2D( - grid, data, theta, num_transit=1, N_B=400, quad=leggauss(3), check=True + grid, + data, + theta, + num_transit=1, + N_B=512, + quad=leggauss(3), + check=True, + warn=False, ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), @@ -1599,7 +1619,7 @@ def test_bounce2d_checks(self): # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) # The result stored at - l, p = 1, 3 + l, p = 0, 3 print("Result(ρ, λ):", result[l, p]) # corresponds to the 1/λ value print("1/λ(ρ):", pitch_inv[l, p]) @@ -1608,9 +1628,9 @@ def test_bounce2d_checks(self): # 10. Plotting fig, ax = bounce.plot_theta(l) - fig, ax = bounce.plot( - l, pitch_inv[l], include_legend=False, show=True, num=10000 - ) + # FIXME: get this plot to match test_bounce1d_checks plot. + # must not be the same part of field line? everything is tested to work... + fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=True) plt.show() return fig diff --git a/tests/test_interp_utils.py b/tests/test_interp_utils.py index 10f8be26c..79a977e1c 100644 --- a/tests/test_interp_utils.py +++ b/tests/test_interp_utils.py @@ -193,19 +193,13 @@ def test_rfftfreq(self, M): @pytest.mark.parametrize( "func, n, domain", [ + # Test cases chosen with purpose, don't remove any. (_f_1d, 2 * _f_1d_nyquist_freq() + 1, (0, 2 * np.pi)), - # At Nyquist: - # Should pass for 1D example above, and useful since - # others don't test if nyquist frequency component is - # computed correctly since it would just be zero. (_f_1d, 2 * _f_1d_nyquist_freq(), (0, 2 * np.pi)), - # shifted domain (_f_1d, 2 * _f_1d_nyquist_freq() + 1, (-np.pi, np.pi)), (_f_1d, 2 * _f_1d_nyquist_freq(), (-np.pi, np.pi)), - # faster period, can do at Nyquist rate if off phase - (lambda x: np.cos(7 * x), 1 * 2, (-np.pi / 7, np.pi / 7)), - # faster period, need above Nyquist rate if on phase - (lambda x: np.sin(7 * x), 1 * 2 + 1, (-np.pi / 7, np.pi / 7)), + (lambda x: np.cos(7 * x), 2, (-np.pi / 7, np.pi / 7)), + (lambda x: np.sin(7 * x), 3, (-np.pi / 7, np.pi / 7)), ], ) def test_interp_rfft(self, func, n, domain): From 3871353fec3c54088bc4bcff72c7f5f1afa463b3 Mon Sep 17 00:00:00 2001 From: unalmis Date: Wed, 2 Oct 2024 23:23:49 -0400 Subject: [PATCH 33/44] Got things working now --- desc/integrals/basis.py | 2 +- desc/integrals/bounce_integral.py | 20 ++-- desc/integrals/bounce_utils.py | 5 +- desc/integrals/interp_utils.py | 10 +- desc/transform.py | 6 +- .../baseline/test_binormal_drift_bounce2d.png | Bin 0 -> 21734 bytes tests/baseline/test_bounce1d_checks.png | Bin 66413 -> 67231 bytes tests/baseline/test_bounce2d_checks.png | Bin 0 -> 68832 bytes tests/test_integrals.py | 87 +++++++----------- tests/test_interp_utils.py | 10 +- 10 files changed, 69 insertions(+), 71 deletions(-) create mode 100644 tests/baseline/test_binormal_drift_bounce2d.png create mode 100644 tests/baseline/test_bounce2d_checks.png diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index a36df24fc..dbcb7372a 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -615,7 +615,7 @@ def check_intersect1d(self, z1, z2, k, plot=True, **kwargs): def plot1d( self, cheb, - num=1000, + num=5000, z1=None, z2=None, k=None, diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 6b8e76987..7aa2d425a 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -394,7 +394,7 @@ def __init__( grid, data, theta, - N_B=32, + N_B=None, num_transit=16, # TODO: Allow multiple starting labels for near-rational surfaces. # think can just concatenate along second to last axis of cheb. @@ -433,6 +433,7 @@ def __init__( ``FourierChebyshevSeries.nodes(M,N,L,domain=(0,2*jnp.pi))``. N_B : int Desired Chebyshev spectral resolution for |B|. + Default is to double the resolution of ``theta``. alpha : float Starting field line poloidal label. num_transit : int @@ -476,6 +477,7 @@ def __init__( check and kwargs.pop("warn", True) and jnp.any(data["B^zeta"] <= 0), msg="(∂ℓ/∂ζ)|ρ,a > 0 is required. Enforcing positive B^ζ.", ) + N_B = setdefault(N_B, theta.shape[-1] * 2) self._alpha = alpha self._m = grid.num_theta self._n = grid.num_zeta @@ -507,7 +509,7 @@ def __init__( assert self._B.N == N_B @staticmethod - def compute_theta(eq, M=32, N=16, rho=1.0, clebsch=None, **kwargs): + def compute_theta(eq, M=16, N=32, rho=1.0, clebsch=None, **kwargs): """Return DESC coordinates θ of Fourier Chebyshev basis nodes. Parameters @@ -765,8 +767,6 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): axes=(-1, -2), ) B = self._B.eval1d(zeta) - # Note not all functions are periodic in ζ with period 2π/NFP. - # Users will likely want GitHub issue #1287. f = [ interp_rfft2( Q, @@ -872,10 +872,16 @@ def plot_theta(self, l, **kwargs): T = self._T if T.cheb.ndim > 2: T = PiecewiseChebyshevSeries(T.cheb[l], T.domain) - kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") - fig, ax = T.plot1d( - T.cheb, **_set_default_plot_kwargs(kwargs, vlabel=r"$\theta$") + kwargs.setdefault( + "title", + r"DESC poloidal angle $\theta($" + + r"$\alpha=$" + + str(self._alpha) + + r"$, \zeta)$", ) + kwargs.setdefault("hlabel", r"$\alpha = $" + str(self._alpha) + r", $\zeta$") + kwargs.setdefault("vlabel", r"$\theta$") + fig, ax = T.plot1d(T.cheb, **_set_default_plot_kwargs(kwargs)) return fig, ax diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 80059891c..d85220a4f 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -234,7 +234,8 @@ def bounce_points( return z1, z2 -def _set_default_plot_kwargs(kwargs, vlabel=r"$\vert B \vert$"): +def _set_default_plot_kwargs(kwargs): + vlabel = r"$\vert B \vert$" kwargs.setdefault( "title", r"Intersects $\zeta$ in epigraph(" @@ -757,7 +758,7 @@ def interp_to_argmin_hard(h, points, knots, g, dg_dz, method="cubic"): def plot_ppoly( ppoly, - num=1000, + num=5000, z1=None, z2=None, k=None, diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index 3c48f3d65..df10490d7 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -24,8 +24,7 @@ # TODO: Transformation to make nodes more uniform Boyd eq. 16.46 pg. 336. -# Have a hunch more uniformly spaced nodes could speed up convergence -# Edit: Seems unnecessary for now. Chebyshev part converges fine. +# More uniformly spaced nodes will speed up convergence. def cheb_pts(N, domain=(-1, 1), lobatto=False): @@ -304,11 +303,12 @@ def irfft2_non_uniform( .divide(1.0 + ((N % 2) == 0)) ) - m = jnp.fft.fftfreq(M, d=np.diff(domain0) / (2 * jnp.pi) / M) - n = jnp.fft.rfftfreq(N, d=np.diff(domain1) / (2 * jnp.pi) / N) idx = np.argsort(axes) - + domain = (domain0, domain1) + m = jnp.fft.fftfreq(M, d=np.diff(domain[idx[0]]) / (2 * jnp.pi) / M) + n = jnp.fft.rfftfreq(N, d=np.diff(domain[idx[1]]) / (2 * jnp.pi) / N) xq = xq - jnp.array([domain0[0], domain1[0]]) + basis = jnp.exp( 1j * ( diff --git a/desc/transform.py b/desc/transform.py index e6cd6f0f3..ff618fd87 100644 --- a/desc/transform.py +++ b/desc/transform.py @@ -73,11 +73,11 @@ def __init__( # and changes variables to the spectrally condensed ζ* = basis.NFP ζ, # so basis.NFP must equal grid.NFP. errorif( - self.grid.NFP != self.basis.NFP + method != "jitable" + and self.grid.NFP != self.basis.NFP and 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dd8b8ca9b..ee8662cba 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1134,11 +1134,12 @@ def test_bounce1d_checks(self): # 1. Define python functions for the integrands. We do that above. # 2. Pick flux surfaces, field lines, and how far to follow the field # line in Clebsch coordinates ρ, α, ζ. - rho = 1 # np.linspace(0.1, 1, 6) - alpha = 0 # np.array([0, 0.5]) - zeta = np.linspace(0, 2 * np.pi, 400) + rho = np.linspace(0.1, 1, 6) + alpha = np.array([0, 0.5]) + zeta = np.linspace(-2 * np.pi, 2 * np.pi, 200) + + eq = get("HELIOTRON") - eq = get("W7-X") # 3. Convert above coordinates to DESC computational coordinates. grid = get_rtz_grid(eq, rho, alpha, zeta, coordinates="raz") # 4. Compute input data. @@ -1146,9 +1147,7 @@ def test_bounce1d_checks(self): Bounce1D.required_names + ["min_tz |B|", "max_tz |B|", "g_zz"], grid=grid ) # 5. Make the bounce integration operator. - bounce = Bounce1D( - grid.source_grid, data, quad=leggauss(3), check=True, warn=False - ) + bounce = Bounce1D(grid.source_grid, data, quad=leggauss(3), check=True) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), max_B=grid.compress(data["max_tz |B|"]), @@ -1186,7 +1185,7 @@ def test_bounce1d_checks(self): # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) # The result stored at - m, l, p = 0, 0, 3 + m, l, p = 0, 1, 3 print("Result(α, ρ, λ):", result[m, l, p]) # corresponds to the 1/λ value print("1/λ(α, ρ):", pitch_inv[l, p]) @@ -1195,7 +1194,7 @@ def test_bounce1d_checks(self): print("(ρ, α):", nodes[m, l, 0]) # 10. Plotting - fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=True) + fig, ax = bounce.plot(m, l, pitch_inv[l], include_legend=False, show=False) return fig @pytest.mark.unit @@ -1421,6 +1420,7 @@ def test_binormal_drift_bounce1d(self): f=f, points=points, check=True, + plot=False, ) drift_numerical_den = bounce.integrate( integrand=TestBounce1D.drift_den_integrand, @@ -1428,6 +1428,7 @@ def test_binormal_drift_bounce1d(self): weight=np.ones(data["zeta"].size), points=points, check=True, + plot=False, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." @@ -1546,6 +1547,7 @@ def _example_numerator(g_zz, B, pitch, zeta): def _example_denominator(B, pitch, zeta): return safediv(1, jnp.sqrt(jnp.abs(1 - pitch * B))) + # TODO: Could test integration against bounce1d for this stellarator @pytest.mark.unit @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d * 4) def test_bounce2d_checks(self): @@ -1559,14 +1561,10 @@ def test_bounce2d_checks(self): # 1. Define python functions for the integrands. We do that above. # 2. Pick flux surfaces and grid resolution. - rho = 1 # np.linspace(0.1, 1, 6) - eq = get("W7-X") + rho = np.linspace(0.1, 1, 6) + eq = get("HELIOTRON") grid = Grid.create_meshgrid( - [ - rho, - fourier_pts(eq._M_grid), - fourier_pts(max(1, eq.N_grid)) / eq.NFP, - ], + [rho, fourier_pts(eq.M_grid), fourier_pts(eq.N_grid) / eq.NFP], period=(np.inf, 2 * np.pi, 2 * np.pi / eq.NFP), NFP=eq.NFP, ) @@ -1574,20 +1572,11 @@ def test_bounce2d_checks(self): data = eq.compute( Bounce2D.required_names + ["min_tz |B|", "max_tz |B|", "g_zz"], grid=grid ) - # 4. Compute DESC coordinates of optimal interpolation nodes. - theta = Bounce2D.compute_theta(eq, M=2, N=512, rho=rho) - + theta = Bounce2D.compute_theta(eq, M=8, N=64, rho=rho) # 5. Make the bounce integration operator. bounce = Bounce2D( - grid, - data, - theta, - num_transit=1, - N_B=512, - quad=leggauss(3), - check=True, - warn=False, + grid, data, theta, num_transit=2, quad=leggauss(3), check=True ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), @@ -1597,7 +1586,7 @@ def test_bounce2d_checks(self): # 6. Compute bounce points. points = bounce.points(pitch_inv) # 7. Optionally check for correctness of bounce points. - bounce.check_points(points, pitch_inv, plot=True) + bounce.check_points(points, pitch_inv, plot=False) # 8. Integrate. num = bounce.integrate( integrand=TestBounce2D._example_numerator, @@ -1613,13 +1602,19 @@ def test_bounce2d_checks(self): check=True, ) avg = safediv(num, den) - assert np.isfinite(avg).all() and np.count_nonzero(avg) + errorif(not np.isfinite(avg).all()) + errorif( + np.count_nonzero(avg) == 0, + msg="Detected 0 wells on this cut of the field line. Make sure enough " + "toroidal transits were followed for this test, or plot the field line " + "to see if this is expected.", + ) # 9. Example manipulation of the output # Sum all bounce averages across a particular field line, for every field line. result = avg.sum(axis=-1) # The result stored at - l, p = 0, 3 + l, p = 1, 3 print("Result(ρ, λ):", result[l, p]) # corresponds to the 1/λ value print("1/λ(ρ):", pitch_inv[l, p]) @@ -1627,11 +1622,7 @@ def test_bounce2d_checks(self): print("ρ:", rho[l]) # 10. Plotting - fig, ax = bounce.plot_theta(l) - # FIXME: get this plot to match test_bounce1d_checks plot. - # must not be the same part of field line? everything is tested to work... - fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=True) - plt.show() + fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) return fig @staticmethod @@ -1646,7 +1637,7 @@ def drift_den_integrand(B, pitch, zeta): return 1 / jnp.sqrt(1 - pitch * B) @pytest.mark.unit - # @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) + @pytest.mark.mpl_image_compare(remove_text=True, tolerance=tol_1d) def test_binormal_drift_bounce2d(self): """Test bounce-averaged drift with analytical expressions.""" data, things = TestBounce1D.get_drift_analytic_data() @@ -1655,9 +1646,6 @@ def test_binormal_drift_bounce2d(self): # Recompute on non-symmetric, fft compatible grid. eq = things["eq"] - # FIXME: Change LinearGrid to default to Fourier points nodes. FFT - # interpolation looks gross on LinearGrid. edit: likely because - # of multi-valued ness issue, but check after done grid = Grid.create_meshgrid( [ data["rho"], @@ -1673,7 +1661,7 @@ def test_binormal_drift_bounce2d(self): grid_data["gbdrift"] = grid_data["gbdrift"] * data["normalization"] # Compute numerical result. - M, N = 16, 16 + M, N = 32, 32 # todo: lower these to minimum once we get match bounce = Bounce2D( grid=grid, data=grid_data, @@ -1684,18 +1672,14 @@ def test_binormal_drift_bounce2d(self): data["rho"], iota=jnp.broadcast_to(data["iota"], shape=(M * N)), ), - N_B=N, - num_transit=3, - # can check if multivalued quantity interpolated correctly - # by comparing plot of f_0 with bounce1d. + num_transit=2, alpha=data["alpha"] - 2 * np.pi * data["iota"], Bref=data["Bref"], Lref=data["a"], check=True, - plot=True, ) points = bounce.points(pitch_inv, num_well=1) - bounce.check_points(points, pitch_inv, plot=True) + bounce.check_points(points, pitch_inv, plot=False) f = Bounce2D.reshape_data(grid, grid_data["cvdrift"], grid_data["gbdrift"]) drift_numerical_num = bounce.integrate( @@ -1704,25 +1688,24 @@ def test_binormal_drift_bounce2d(self): f=f, points=points, check=True, - plot=True, + plot=False, ) drift_numerical_den = bounce.integrate( integrand=TestBounce2D.drift_den_integrand, pitch_inv=pitch_inv, points=points, check=True, - plot=True, + plot=False, ) drift_numerical = np.squeeze(drift_numerical_num / drift_numerical_den) msg = "There should be one bounce integral per pitch in this example." assert drift_numerical.size == drift_analytic.size, msg fig, ax = plt.subplots() - ax.plot(pitch_inv, drift_analytic, label="analytic") - ax.plot(pitch_inv, drift_numerical, label="numerical") - plt.legend() - plt.show() + ax.plot(pitch_inv, drift_analytic) + ax.plot(pitch_inv, drift_numerical) + # TODO: need to make integrands multivalued # np.testing.assert_allclose( # noqa: E800 # drift_numerical, drift_analytic, atol=5e-3, rtol=5e-2 # noqa: E800 # ) # noqa: E800 diff --git a/tests/test_interp_utils.py b/tests/test_interp_utils.py index 79a977e1c..fcc0ea90e 100644 --- a/tests/test_interp_utils.py +++ b/tests/test_interp_utils.py @@ -218,6 +218,7 @@ def test_interp_rfft(self, func, n, domain): @pytest.mark.parametrize( "func, m, n, domain0, domain1", [ + # Test cases chosen with purpose, don't remove any. ( _f_2d, 2 * _f_2d_nyquist_freq()[0] + 1, @@ -232,11 +233,18 @@ def test_interp_rfft(self, func, n, domain): (-np.pi / 3, 5 * np.pi / 3), (np.pi, 3 * np.pi), ), + ( + lambda x, y: np.cos(30 * x) + np.sin(y) ** 2 + 1, + 2 * 30 // 30 + 1, + 2 * 2 + 1, + (0, 2 * np.pi / 30), + (np.pi, 3 * np.pi), + ), ], ) def test_interp_rfft2(self, func, m, n, domain0, domain1): """Test non-uniform FFT interpolation.""" - xq = np.array([[7.34, 1.10134, 2.28], [1.1, 3.78432, 8.542]]).T + xq = np.array([[7.34, 1.10134, 2.28, 1e3 * np.e], [1.1, 3.78432, 8.542, 0]]).T x = np.linspace(domain0[0], domain0[1], m, endpoint=False) y = np.linspace(domain1[0], domain1[1], n, endpoint=False) x, y = map(np.ravel, list(np.meshgrid(x, y, indexing="ij"))) From 60fc00f690b45cb6f51f773e13c86cc97b317dd4 Mon Sep 17 00:00:00 2001 From: unalmis Date: Thu, 3 Oct 2024 00:14:20 -0400 Subject: [PATCH 34/44] Change NFP mismatch error back to warning --- desc/equilibrium/coords.py | 1 - desc/transform.py | 6 +++--- 2 files changed, 3 insertions(+), 4 deletions(-) diff --git a/desc/equilibrium/coords.py b/desc/equilibrium/coords.py index ebf78e622..7e50a71a6 100644 --- a/desc/equilibrium/coords.py +++ b/desc/equilibrium/coords.py @@ -740,7 +740,6 @@ def get_rtz_grid( desc_grid = Grid( nodes=rtz_nodes, coordinates="rtz", - NFP=1, source_grid=grid, sort=False, jitable=jitable, diff --git a/desc/transform.py b/desc/transform.py index ff618fd87..2ab54ccc9 100644 --- a/desc/transform.py +++ b/desc/transform.py @@ -10,10 +10,10 @@ from desc.io import IOAble from desc.utils import ( combination_permutation, - errorif, isalmostequal, islinspaced, issorted, + warnif, ) @@ -65,14 +65,14 @@ def __init__( self._basis = basis self._rcond = rcond if rcond is not None else "auto" - errorif( + warnif( self.grid.coordinates != "rtz", msg=f"Expected coordinates rtz got {self.grid.coordinates}.", ) # DESC truncates the computational domain to ζ ∈ [0, 2π/grid.NFP] # and changes variables to the spectrally condensed ζ* = basis.NFP ζ, # so basis.NFP must equal grid.NFP. - errorif( + warnif( method != "jitable" and self.grid.NFP != self.basis.NFP and self.basis.N != 0 From 56dcbb27ff269d22a134857fdd8cf894cf70bd2d Mon Sep 17 00:00:00 2001 From: unalmis Date: Sat, 5 Oct 2024 21:36:34 -0400 Subject: [PATCH 35/44] Split compute quantities into periodic and secular terms --- desc/compute/_basis_vectors.py | 50 ++++++++++++- desc/compute/_core.py | 46 ++++++++++-- desc/compute/_metric.py | 90 ++++++++++++++++++++--- tests/inputs/master_compute_data_rpz.pkl | Bin 8677545 -> 8730763 bytes tests/test_integrals.py | 22 +++--- 5 files changed, 181 insertions(+), 27 deletions(-) diff --git a/desc/compute/_basis_vectors.py b/desc/compute/_basis_vectors.py index 72803a1d7..4787eb860 100644 --- a/desc/compute/_basis_vectors.py +++ b/desc/compute/_basis_vectors.py @@ -3223,17 +3223,61 @@ def _e_sub_zeta_zz(params, transforms, profiles, data, **kwargs): transforms={}, profiles=[], coordinates="rtz", - data=["e^rho", "e^theta", "e^zeta", "alpha_r", "alpha_t", "alpha_z"], + data=["periodic(grad(alpha))", "secular(grad(alpha))"], ) def _grad_alpha(params, transforms, profiles, data, **kwargs): - data["grad(alpha)"] = ( - data["alpha_r"] * data["e^rho"].T + data["grad(alpha)"] = data["periodic(grad(alpha))"] + data["secular(grad(alpha))"] + return data + + +@register_compute_fun( + name="periodic(grad(alpha))", + label="\\mathrm{periodic}(\\nabla \\alpha)", + units="m^{-1}", + units_long="Inverse meters", + description=( + "Gradient of field line label, which is perpendicular to the field line, " + "periodic component" + ), + dim=3, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["e^rho", "e^theta", "e^zeta", "periodic(alpha_r)", "alpha_t", "alpha_z"], +) +def _periodic_grad_alpha(params, transforms, profiles, data, **kwargs): + data["periodic(grad(alpha))"] = ( + data["periodic(alpha_r)"] * data["e^rho"].T + data["alpha_t"] * data["e^theta"].T + data["alpha_z"] * data["e^zeta"].T ).T return data +@register_compute_fun( + name="secular(grad(alpha))", + label="\\mathrm{secular}(\\nabla \\alpha)", + units="m^{-1}", + units_long="Inverse meters", + description=( + "Gradient of field line label, which is perpendicular to the field line, " + "periodic component" + ), + dim=3, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["e^rho", "secular(alpha_r)"], +) +def _secular_grad_alpha(params, transforms, profiles, data, **kwargs): + data["secular(grad(alpha))"] = ( + data["secular(alpha_r)"][:, jnp.newaxis] * data["e^rho"] + ) + return data + + @register_compute_fun( name="grad(psi)", label="\\nabla\\psi", diff --git a/desc/compute/_core.py b/desc/compute/_core.py index 784e296f7..ef0466f8b 100644 --- a/desc/compute/_core.py +++ b/desc/compute/_core.py @@ -1518,14 +1518,48 @@ def _alpha(params, transforms, profiles, data, **kwargs): transforms={}, profiles=[], coordinates="rtz", - data=["theta_PEST_r", "phi", "phi_r", "iota", "iota_r"], + data=["periodic(alpha_r)", "secular(alpha_r)"], ) def _alpha_r(params, transforms, profiles, data, **kwargs): - data["alpha_r"] = ( - data["theta_PEST_r"] - - data["iota_r"] * data["phi"] - - data["iota"] * data["phi_r"] - ) + data["alpha_r"] = data["periodic(alpha_r)"] + data["secular(alpha_r)"] + return data + + +@register_compute_fun( + name="periodic(alpha_r)", + label="\\mathrm{periodic}(\\partial_\\rho \\alpha)", + units="~", + units_long="None", + description="Field line label, derivative wrt radial coordinate, " + "periodic component", + dim=1, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["theta_PEST_r", "iota", "phi_r"], +) +def _periodic_alpha_r(params, transforms, profiles, data, **kwargs): + data["periodic(alpha_r)"] = data["theta_PEST_r"] - data["iota"] * data["phi_r"] + return data + + +@register_compute_fun( + name="secular(alpha_r)", + label="\\mathrm{secular}(\\partial_\\rho \\alpha)", + units="~", + units_long="None", + description="Field line label, derivative wrt radial coordinate, " + "secular component", + dim=1, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["iota_r", "phi"], +) +def _secular_alpha_r(params, transforms, profiles, data, **kwargs): + data["secular(alpha_r)"] = -data["iota_r"] * data["phi"] return data diff --git a/desc/compute/_metric.py b/desc/compute/_metric.py index da4353ad0..0d66b8305 100644 --- a/desc/compute/_metric.py +++ b/desc/compute/_metric.py @@ -1941,8 +1941,8 @@ def _g_sup_ra(params, transforms, profiles, data, **kwargs): # Exact definition of the magnetic drifts taken from # eqn. 48 of Introduction to Quasisymmetry by Landreman # https://tinyurl.com/54udvaa4 - label="\\mathrm{gbdrift} = 1/B^{2} (\\mathbf{b}\\times\\nabla B) \\cdot" - + "\\nabla \\alpha", + label="(\\nabla \\vert B \\vert)_{\\mathrm{drift}} = " + "(\\mathbf{b} \\times \\nabla B) \\cdot \\nabla \\alpha / \\vert B \\vert^{2}", units="1/(T-m^{2})", units_long="inverse Tesla meters^2", description="Binormal component of the geometric part of the gradB drift" @@ -1952,13 +1952,61 @@ def _g_sup_ra(params, transforms, profiles, data, **kwargs): transforms={}, profiles=[], coordinates="rtz", - data=["|B|^2", "b", "grad(alpha)", "grad(|B|)"], + data=["periodic(gbdrift)", "secular(gbdrift)"], ) def _gbdrift(params, transforms, profiles, data, **kwargs): - data["gbdrift"] = ( - 1 + data["gbdrift"] = data["periodic(gbdrift)"] + data["secular(gbdrift)"] + return data + + +@register_compute_fun( + name="periodic(gbdrift)", + # Exact definition of the magnetic drifts taken from + # eqn. 48 of Introduction to Quasisymmetry by Landreman + # https://tinyurl.com/54udvaa4 + label="\\mathrm{periodic}(\\nabla \\vert B \\vert)_{\\mathrm{drift}}", + units="1/(T-m^{2})", + units_long="inverse Tesla meters^2", + description="Binormal component of the geometric part of the gradB drift" + + " used for local stability analyses, Gamma_c, epsilon_eff etc." + " Periodic component.", + dim=1, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["|B|^2", "b", "periodic(grad(alpha))", "grad(|B|)"], +) +def _periodic_gbdrift(params, transforms, profiles, data, **kwargs): + data["periodic(gbdrift)"] = ( + dot(data["b"], cross(data["grad(|B|)"], data["periodic(grad(alpha))"])) + / data["|B|^2"] + ) + return data + + +@register_compute_fun( + name="secular(gbdrift)", + # Exact definition of the magnetic drifts taken from + # eqn. 48 of Introduction to Quasisymmetry by Landreman + # https://tinyurl.com/54udvaa4 + label="\\mathrm{secular}(\\nabla \\vert B \\vert)_{\\mathrm{drift}}", + units="1/(T-m^{2})", + units_long="inverse Tesla meters^2", + description="Binormal component of the geometric part of the gradB drift" + + " used for local stability analyses, Gamma_c, epsilon_eff etc. " + "Secular component.", + dim=1, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["|B|^2", "b", "secular(grad(alpha))", "grad(|B|)"], +) +def _secular_gbdrift(params, transforms, profiles, data, **kwargs): + data["secular(gbdrift)"] = ( + dot(data["b"], cross(data["grad(|B|)"], data["secular(grad(alpha))"])) / data["|B|^2"] - * dot(data["b"], cross(data["grad(|B|)"], data["grad(alpha)"])) ) return data @@ -1979,11 +2027,35 @@ def _gbdrift(params, transforms, profiles, data, **kwargs): transforms={}, profiles=[], coordinates="rtz", - data=["p_r", "psi_r", "|B|^2", "gbdrift"], + data=["periodic(cvdrift)", "secular(gbdrift)"], ) def _cvdrift(params, transforms, profiles, data, **kwargs): - dp_dpsi = mu_0 * data["p_r"] / data["psi_r"] - data["cvdrift"] = 1 / data["|B|^2"] * dp_dpsi + data["gbdrift"] + data["cvdrift"] = data["periodic(cvdrift)"] + data["secular(gbdrift)"] + return data + + +@register_compute_fun( + name="periodic(cvdrift)", + # Exact definition of the magnetic drifts taken from + # eqn. 48 of Introduction to Quasisymmetry by Landreman + # https://tinyurl.com/54udvaa4 + label="\\mathrm{periodic(cvdrift)}", + units="1/(T-m^{2})", + units_long="inverse Tesla meters^2", + description="Binormal component of the geometric part of the curvature drift" + + " used for local stability analyses, Gamma_c, epsilon_eff etc. " + "Periodic component.", + dim=1, + params=[], + transforms={}, + profiles=[], + coordinates="rtz", + data=["p_r", "psi_r", "|B|^2", "periodic(gbdrift)"], +) +def _periodic_cvdrift(params, transforms, profiles, data, **kwargs): + data["periodic(cvdrift)"] = ( + mu_0 * data["p_r"] / data["psi_r"] / data["|B|^2"] + data["periodic(gbdrift)"] + ) return data diff --git a/tests/inputs/master_compute_data_rpz.pkl b/tests/inputs/master_compute_data_rpz.pkl index 7a887be3de2790dcbf3bb333b9594fbfa803c7d8..ad8205555e2636aa2dc51e1140fcaf2d44999e78 100644 GIT binary patch delta 3589835 zcmeFac|28J|Nn10X36XrDno|MQ<;*rh@@1Ml2oS9tTa%nV<>4BvKrA$8k9n$wG|mM zh0K|!kRgOjzjNK!<@34k`?|WE-{bN7{yz8T@jd?J*~ZW`^0^^%yzCn zf{o3e?aIo_Y=)d1_D*I8y}U0kkCmwTxbo@Fw_&Cc2P2M0oagp&ciH81@W?**i12xA 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z7{;07re=Vq__9;Q5I5$8?#swZU`pU0d6PK~GKJ2v?Lz16@FbUWvvnnyNN=|9fngV! z<u}oM5Es1zzHggwR2FzWj|3ZhzDmMg*_bVwCRmYr)e>4tufySXPF2u}=o$&fWMk%C zM%QFrBxB|^dmFGC$(VUfrADlVzx|ZJjB*Ds{LQqQK$lLE=Y=V(ao;8hS+e{M6_V5q z6_TtuR>SHf9AmPs?2+Jtw^xPn(j8%FJtIDyR%4C|br6x`4?_*WBr&(H3mh?eN&Q1h zQOp4MmTa~h%pXn(4HbC5C%l4{Y?tG4+!M%>ZfJ1e=jtb9G3f$;i|XW3xRU?gOHr_A oXsd<qxB>+rSpx-<J@e_pn*-KNyc`k9o;huj74zA1TP5WEAAOeZ3IG5A diff --git a/tests/test_integrals.py b/tests/test_integrals.py index ee8662cba..0aabb9fbe 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1261,6 +1261,7 @@ def get_drift_analytic_data(): "iota", "psi", "a", + "theta_PEST", ], grid=grid, ) @@ -1303,15 +1304,14 @@ def drift_analytic(data): # is independent of normalization length scales, like "effective r/R0". epsilon = data["a"] * data["rho"] # Aspect ratio of the flux surface. np.testing.assert_allclose(epsilon, 0.05) - theta_PEST = data["alpha"] + data["iota"] * data["zeta"] # same as 1 / (1 + epsilon cos(theta)) assuming epsilon << 1 - B_analytic = B0 * (1 - epsilon * np.cos(theta_PEST)) + B_analytic = B0 * (1 - epsilon * np.cos(data["theta_PEST"])) np.testing.assert_allclose(B, B_analytic, atol=3e-3) gradpar = data["a"] * data["B^zeta"] / data["|B|"] # This method of computing G0 suggests a fixed point iteration. G0 = data["a"] - gradpar_analytic = G0 * (1 - epsilon * np.cos(theta_PEST)) + gradpar_analytic = G0 * (1 - epsilon * np.cos(data["theta_PEST"])) gradpar_theta_analytic = data["iota"] * gradpar_analytic G0 = np.mean(gradpar_theta_analytic) np.testing.assert_allclose(gradpar, gradpar_analytic, atol=5e-3) @@ -1329,10 +1329,12 @@ def drift_analytic(data): / data["Bref"] ) gds21_analytic = -data["shear"] * ( - data["shear"] * theta_PEST - alpha_MHD / B**4 * np.sin(theta_PEST) + data["shear"] * data["theta_PEST"] + - alpha_MHD / B**4 * np.sin(data["theta_PEST"]) ) gds21_analytic_low_order = -data["shear"] * ( - data["shear"] * theta_PEST - alpha_MHD / B0**4 * np.sin(theta_PEST) + data["shear"] * data["theta_PEST"] + - alpha_MHD / B0**4 * np.sin(data["theta_PEST"]) ) np.testing.assert_allclose(gds21, gds21_analytic, atol=2e-2) np.testing.assert_allclose(gds21, gds21_analytic_low_order, atol=2.7e-2) @@ -1340,13 +1342,13 @@ def drift_analytic(data): fudge_1 = 0.19 gbdrift_analytic = fudge_1 * ( -data["shear"] - + np.cos(theta_PEST) - - gds21_analytic / data["shear"] * np.sin(theta_PEST) + + np.cos(data["theta_PEST"]) + - gds21_analytic / data["shear"] * np.sin(data["theta_PEST"]) ) gbdrift_analytic_low_order = fudge_1 * ( -data["shear"] - + np.cos(theta_PEST) - - gds21_analytic_low_order / data["shear"] * np.sin(theta_PEST) + + np.cos(data["theta_PEST"]) + - gds21_analytic_low_order / data["shear"] * np.sin(data["theta_PEST"]) ) fudge_2 = 0.07 cvdrift_analytic = gbdrift_analytic + fudge_2 * alpha_MHD / B**2 @@ -1654,6 +1656,8 @@ def test_binormal_drift_bounce2d(self): ], NFP=eq.NFP, ) + # TODO: request periodic terms and construct secular terms in the + # integrand functions grid_data = eq.compute( names=Bounce2D.required_names + ["cvdrift", "gbdrift"], grid=grid ) From 30d511e3c93431dfc16262514903933a8419089b Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 00:24:34 -0400 Subject: [PATCH 36/44] Add ability to compute length along field line with fft --- desc/integrals/basis.py | 2 +- desc/integrals/bounce_integral.py | 83 ++++++++++++++++++++++++------- desc/integrals/interp_utils.py | 63 ++++++++++++----------- tests/test_integrals.py | 20 ++++++++ tests/test_interp_utils.py | 9 ++-- 5 files changed, 125 insertions(+), 52 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index dbcb7372a..8369ce583 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -372,7 +372,7 @@ def eval1d(self, z, cheb=None): ---------- z : jnp.ndarray Shape (..., *cheb.shape[:-2], z.shape[-1]). - Coordinates in [sef.domain[0], ∞). + Coordinates in [self.domain[0], ∞). The coordinates z ∈ ℝ are assumed isomorphic to (x, y) ∈ ℝ² where ``z // domain`` yields the index into the proper Chebyshev series along the second to last axis of ``cheb`` and ``z % domain`` is diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 7aa2d425a..dadde2662 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -20,6 +20,7 @@ from desc.integrals.interp_utils import ( cheb_from_dct, cheb_pts, + idct_non_uniform, interp_rfft2, irfft2_non_uniform, polyder_vec, @@ -92,15 +93,17 @@ def _transform_to_clebsch(grid, nodes, f, is_reshaped=False): f = grid.meshgrid_reshape(f, "rtz") M, N = nodes.shape[-3], nodes.shape[-2] + nodes = nodes.reshape(*nodes.shape[:-3], M * N, 2) return FourierChebyshevSeries( f=interp_rfft2( # Interpolate to nodes in Clebsch space, # which is not a tensor product node set in DESC space. - xq=nodes.reshape(*nodes.shape[:-3], M * N, 2), + xq0=nodes[..., 0], + xq1=nodes[..., 1], f=f[..., jnp.newaxis, :, :], domain1=(0, 2 * jnp.pi / grid.NFP), axes=(-1, -2), - ).reshape(*nodes.shape[:-3], M, N), + ).reshape(*nodes.shape[:-2], M, N), domain=(0, 2 * jnp.pi), ) @@ -185,12 +188,12 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): T = FourierChebyshevSeries(f=theta, domain=(0, 2 * jnp.pi)).compute_cheb(alpha) T.stitch() theta = T.evaluate(N_B) - xq = jnp.stack( - [theta, jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape)], axis=-1 - ).reshape(*alpha.shape[:-1], alpha.shape[-1] * N_B, 2) + zeta = jnp.broadcast_to(cheb_pts(N_B, domain=T.domain), theta.shape) + shape = (*alpha.shape[:-1], alpha.shape[-1] * N_B) B = interp_rfft2( - xq=xq, + theta.reshape(shape), + zeta.reshape(shape), f=B[..., jnp.newaxis, :, :], domain1=(0, 2 * jnp.pi / grid.NFP), axes=(-1, -2), @@ -501,8 +504,8 @@ def __init__( self._T, self._B = _transform_to_clebsch_1d( grid, alpha, theta, data["|B|"] / Bref, N_B, is_reshaped ) - self._b_sup_z = _transform_to_desc( - grid, jnp.abs(data["B^zeta"]) / data["|B|"] * Lref, is_reshaped + self._B_sup_z = _transform_to_desc( + grid, jnp.abs(data["B^zeta"]) * Lref / Bref, is_reshaped ) assert self._T.M == self._B.M == num_transit assert self._T.N == theta.shape[-1] @@ -743,24 +746,26 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): Shape (num_pitch, ) or (num_pitch, L). f : list[jnp.ndarray] Shape (m, n) or (L, m, n). - f : list[jnp.ndarray] + f_vec : list[jnp.ndarray] Shape (m, n, 3) or (L, m, n, 3). """ z1, z2 = points shape = [*z1.shape, self._x.size] - # This is ζ along the field line. + # These are the ζ coordinates of the quadrature points. + # Shape is (num_pitch, L, number of points to interpolate onto). zeta = flatten_matrix( bijection_from_disc(self._x, z1[..., jnp.newaxis], z2[..., jnp.newaxis]) ) # Note self._T expects shape (num_pitch, L) if T.cheb.shape[0] is L. - # These are the (θ, ζ) coordinates of the quadrature points. - Q = jnp.stack([self._T.eval1d(zeta), zeta], axis=-1) + # These are the θ coordinates of the quadrature points. + theta = self._T.eval1d(zeta) - b_sup_z = irfft2_non_uniform( - xq=Q, - a=self._b_sup_z[..., jnp.newaxis, :, :], + B_sup_z = irfft2_non_uniform( + theta, + zeta, + a=self._B_sup_z[..., jnp.newaxis, :, :], M=self._n, N=self._m, domain1=(0, 2 * jnp.pi / self._NFP), @@ -769,7 +774,8 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): B = self._B.eval1d(zeta) f = [ interp_rfft2( - Q, + theta, + zeta, f_i[..., jnp.newaxis, :, :], domain1=(0, 2 * jnp.pi / self._NFP), axes=(-1, -2), @@ -778,7 +784,8 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): ] f_vec = [ interp_rfft2( - Q[..., jnp.newaxis, :], + theta[..., jnp.newaxis], + zeta[..., jnp.newaxis], f_i[..., jnp.newaxis, :, :, :], domain1=(0, 2 * jnp.pi / self._NFP), axes=(-2, -3), @@ -794,7 +801,8 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): pitch=1 / pitch_inv[..., jnp.newaxis], zeta=zeta, ) - / b_sup_z + * B + / B_sup_z ) .reshape(shape) .dot(self._w) @@ -806,13 +814,50 @@ def _integrate(self, integrand, points, pitch_inv, f, f_vec, check, plot): _check_interp( # num_alpha is 1, num_rho, num_pitch, num_well, num_quad (1, *shape), - *map(_swap_pl, (zeta, b_sup_z, B)), + *map(_swap_pl, (zeta, B_sup_z, B)), result, list(map(_swap_pl, f)), plot, ) return result + def compute_length(self, quad=None): + """Compute the proper length of the field line ∫ dℓ / |B|. + + Parameters + ---------- + quad : tuple[jnp.ndarray] + Quadrature points xₖ and weights wₖ for the approximate evaluation + of the integral ∫₋₁¹ f(x) dx ≈ ∑ₖ wₖ f(xₖ). + Should not use more points than half Chebyshev resolution of |B|. + + Returns + ------- + length : jnp.ndarray + Shape (L, ). + + """ + # Gauss quadrature captures double frequency of Chebyshev series. + x, w = leggauss(self._B.N // 2) if quad is None else quad + + # TODO: There exits a fast transform from Chebyshev series to Legendre nodes. + theta = idct_non_uniform(x, self._T.cheb[..., jnp.newaxis, :], self._T.N) + zeta = jnp.broadcast_to(bijection_from_disc(x, 0, 2 * jnp.pi), theta.shape) + + shape = (-1, self._T.M * w.size) # (num_rho, num transit * w.size) + B_sup_z = irfft2_non_uniform( + theta.reshape(shape), + zeta.reshape(shape), + a=self._B_sup_z[..., jnp.newaxis, :, :], + M=self._n, + N=self._m, + domain1=(0, 2 * jnp.pi / self._NFP), + axes=(-1, -2), + ).reshape(-1, self._T.M, w.size) + + # Gradient of change of variable bijection from [−1, 1] → [0, 2π] is π. + return (1 / B_sup_z).dot(w).sum(axis=-1) * jnp.pi + def plot(self, l, pitch_inv=None, **kwargs): """Plot the field line and bounce points of the given pitch angles. diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index df10490d7..b600c6f84 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -218,26 +218,29 @@ def irfft_non_uniform(xq, a, n, domain=(0, 2 * jnp.pi), axis=-1): def interp_rfft2( - xq, f, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) + xq0, xq1, f, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) ): - """Interpolate real-valued ``f`` to ``xq`` with FFT. + """Interpolate real-valued ``f`` to coordinates ``(xq0,xq1)`` with FFT. Parameters ---------- - xq : jnp.ndarray - Shape (..., 2). - Real query points where interpolation is desired. - Shape ``xq.shape[:-1]`` must broadcast with shape ``np.delete(f.shape,axes)``. - Last axis must hold coordinates for a given point. The coordinates stored - along ``xq[...,0]`` (``xq[...,1]``) must be the same coordinate enumerated - across axis ``min(axes)`` (``max(axes)``) of the function values ``f``. + xq0 : jnp.ndarray + Real query points of coordinate in ``domain0`` where interpolation is desired. + Shape must broadcast with shape ``np.delete(a.shape,axes)``. + The coordinates stored here must be the same coordinate enumerated + across axis ``min(axes)`` of the function values ``f``. + xq1 : jnp.ndarray + Real query points of coordinate in ``domain1`` where interpolation is desired. + Shape must broadcast with shape ``np.delete(a.shape,axes)``. + The coordinates stored here must be the same coordinate enumerated + across axis ``max(axes)`` of the function values ``f``. f : jnp.ndarray Shape (..., f.shape[-2], f.shape[-1]). Real function values on uniform tensor-product grid over an open period. domain0 : tuple[float] - Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. + Domain of coordinate specified by ``xq0`` over which samples were taken. domain1 : tuple[float] - Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. + Domain of coordinate specified by ``xq1`` over which samples were taken. axes : tuple[int] Axes along which to transform. The real transform is done along ``axes[1]``, so it will be more @@ -251,26 +254,28 @@ def interp_rfft2( """ a = rfft2(f, axes=axes, norm="forward") fq = irfft2_non_uniform( - xq, a, f.shape[axes[0]], f.shape[axes[1]], domain0, domain1, axes + xq0, xq1, a, f.shape[axes[0]], f.shape[axes[1]], domain0, domain1, axes ) return fq def irfft2_non_uniform( - xq, a, M, N, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) + xq0, xq1, a, M, N, domain0=(0, 2 * jnp.pi), domain1=(0, 2 * jnp.pi), axes=(-2, -1) ): - """Evaluate Fourier coefficients ``a`` at ``xq``. + """Evaluate Fourier coefficients ``a`` at coordinates ``(xq0,xq1)``. Parameters ---------- - xq : jnp.ndarray - Shape (..., 2). - Real query points where interpolation is desired. - Last axis must hold coordinates for a given point. - Shape ``xq.shape[:-1]`` must broadcast with shape ``np.delete(a.shape,axes)``. - Last axis must hold coordinates for a given point. The coordinates stored - along ``xq[...,0]`` (``xq[...,1]``) must be the same coordinate enumerated - across axis ``min(axes)`` (``max(axes)``) of the Fourier coefficients ``a``. + xq0 : jnp.ndarray + Real query points of coordinate in ``domain0`` where interpolation is desired. + Shape must broadcast with shape ``np.delete(a.shape,axes)``. + The coordinates stored here must be the same coordinate enumerated + across axis ``min(axes)`` of the Fourier coefficients ``a``. + xq1 : jnp.ndarray + Real query points of coordinate in ``domain1`` where interpolation is desired. + Shape must broadcast with shape ``np.delete(a.shape,axes)``. + The coordinates stored here must be the same coordinate enumerated + across axis ``max(axes)`` of the Fourier coefficients ``a``. a : jnp.ndarray Shape (..., a.shape[-2], a.shape[-1]). Fourier coefficients ``a=rfft2(f,axes=axes,norm="forward")``. @@ -279,9 +284,9 @@ def irfft2_non_uniform( N : int Spectral resolution of ``a`` along ``axes[1]``. domain0 : tuple[float] - Domain of coordinate specified by ``xq[...,0]`` over which samples were taken. + Domain of coordinate specified by ``xq0`` over which samples were taken. domain1 : tuple[float] - Domain of coordinate specified by ``xq[...,1]`` over which samples were taken. + Domain of coordinate specified by ``xq1`` over which samples were taken. axes : tuple[int] Axes along which to transform. @@ -291,7 +296,7 @@ def irfft2_non_uniform( Real function value at query points. """ - errorif(not (len(axes) == xq.shape[-1] == 2), msg="This is a 2D transform.") + errorif(len(axes) != 2, msg="This is a 2D transform.") errorif(a.ndim < 2, msg=f"Dimension mismatch, a.shape: {a.shape}.") # |a| << |basis|, so move a instead of basis @@ -307,13 +312,15 @@ def irfft2_non_uniform( domain = (domain0, domain1) m = jnp.fft.fftfreq(M, d=np.diff(domain[idx[0]]) / (2 * jnp.pi) / M) n = jnp.fft.rfftfreq(N, d=np.diff(domain[idx[1]]) / (2 * jnp.pi) / N) - xq = xq - jnp.array([domain0[0], domain1[0]]) + xq0 = xq0 - domain0[0] + xq1 = xq1 - domain1[0] + xq = (xq0, xq1) basis = jnp.exp( 1j * ( - (m * xq[..., idx[0], jnp.newaxis])[..., jnp.newaxis] - + (n * xq[..., idx[1], jnp.newaxis])[..., jnp.newaxis, :] + (m * xq[idx[0]][..., jnp.newaxis])[..., jnp.newaxis] + + (n * xq[idx[1]][..., jnp.newaxis])[..., jnp.newaxis, :] ) ) fq = 2.0 * (basis * a).real.sum(axis=(-2, -1)) diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 0aabb9fbe..a04b391c2 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1625,6 +1625,26 @@ def test_bounce2d_checks(self): # 10. Plotting fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) + + length = bounce.compute_length() + np.testing.assert_allclose( + length, + # Computed through <L|r,a> with Simpson's rule at 800 nodes (over-resolved). + # The difference is likely not due to quadrature error, rather interpolation + # error as the data points for ``bounce.compute_length()`` come from Fourier + # series of |B|, while those for <L|r,a> come from Fourier series of + # plasma boundary. Also, currently JAX has a bug with DCT and FFT that limit + # the accuracy to something comparable to 32 bit. + [ + 384.77892007, + 361.60220181, + 345.33817065, + 333.00781712, + 352.16277188, + 440.09424799, + ], + rtol=3e-3, + ) return fig @staticmethod diff --git a/tests/test_interp_utils.py b/tests/test_interp_utils.py index fcc0ea90e..e0843acd7 100644 --- a/tests/test_interp_utils.py +++ b/tests/test_interp_utils.py @@ -244,18 +244,19 @@ def test_interp_rfft(self, func, n, domain): ) def test_interp_rfft2(self, func, m, n, domain0, domain1): """Test non-uniform FFT interpolation.""" - xq = np.array([[7.34, 1.10134, 2.28, 1e3 * np.e], [1.1, 3.78432, 8.542, 0]]).T + theta = np.array([7.34, 1.10134, 2.28, 1e3 * np.e]) + zeta = np.array([1.1, 3.78432, 8.542, 0]) x = np.linspace(domain0[0], domain0[1], m, endpoint=False) y = np.linspace(domain1[0], domain1[1], n, endpoint=False) x, y = map(np.ravel, list(np.meshgrid(x, y, indexing="ij"))) - truth = func(xq[..., 0], xq[..., 1]) + truth = func(theta, zeta) f = func(x, y).reshape(m, n) np.testing.assert_allclose( - interp_rfft2(xq, f, domain0, domain1, axes=(-2, -1)), + interp_rfft2(theta, zeta, f, domain0, domain1, axes=(-2, -1)), truth, ) np.testing.assert_allclose( - interp_rfft2(xq, f, domain0, domain1, axes=(-1, -2)), + interp_rfft2(theta, zeta, f, domain0, domain1, axes=(-1, -2)), truth, ) From 6870bcd8298a4c4cd8932899721aff430b604d85 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 00:33:40 -0400 Subject: [PATCH 37/44] Make sure returned shape doesn't have extra dimension --- desc/integrals/bounce_integral.py | 2 +- tests/test_integrals.py | 19 ++++++++----------- 2 files changed, 9 insertions(+), 12 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index dadde2662..679c2f9d1 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -853,7 +853,7 @@ def compute_length(self, quad=None): N=self._m, domain1=(0, 2 * jnp.pi / self._NFP), axes=(-1, -2), - ).reshape(-1, self._T.M, w.size) + ).reshape(*self._T.cheb.shape[:-2], self._T.M, w.size) # Gradient of change of variable bijection from [−1, 1] → [0, 2π] is π. return (1 / B_sup_z).dot(w).sum(axis=-1) * jnp.pi diff --git a/tests/test_integrals.py b/tests/test_integrals.py index a04b391c2..9de919126 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1623,18 +1623,12 @@ def test_bounce2d_checks(self): # for the flux surface print("ρ:", rho[l]) - # 10. Plotting - fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) - - length = bounce.compute_length() np.testing.assert_allclose( - length, - # Computed through <L|r,a> with Simpson's rule at 800 nodes (over-resolved). - # The difference is likely not due to quadrature error, rather interpolation - # error as the data points for ``bounce.compute_length()`` come from Fourier - # series of |B|, while those for <L|r,a> come from Fourier series of - # plasma boundary. Also, currently JAX has a bug with DCT and FFT that limit - # the accuracy to something comparable to 32 bit. + bounce.compute_length(), + # Computed data below through <L|r,a> with Simpson's rule at 800 nodes. + # The difference is likely due to interpolation and floating point error. + # (On the version of JAX on which rtol was set, there is a bug with DCT + # and FFT that limit the accuracy to something comparable to 32 bit). [ 384.77892007, 361.60220181, @@ -1645,6 +1639,9 @@ def test_bounce2d_checks(self): ], rtol=3e-3, ) + + # 10. Plotting + fig, ax = bounce.plot(l, pitch_inv[l], include_legend=False, show=False) return fig @staticmethod From 5e725937f97d185920413a51abaa7e0aee377680 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 01:49:21 -0400 Subject: [PATCH 38/44] PUlling changes from downstream branch --- desc/integrals/basis.py | 3 +- desc/integrals/bounce_integral.py | 52 ++++++++++++++----------------- desc/integrals/bounce_utils.py | 3 +- tests/test_integrals.py | 11 +++++-- 4 files changed, 36 insertions(+), 33 deletions(-) diff --git a/desc/integrals/basis.py b/desc/integrals/basis.py index 8369ce583..18379fc34 100644 --- a/desc/integrals/basis.py +++ b/desc/integrals/basis.py @@ -208,13 +208,12 @@ def evaluate(self, M, N): ``FourierChebyshevSeries.nodes(M,N,L,self.domain,self.lobatto)``. """ - fq = idct( + return idct( irfft(self._c, n=M, axis=-2, norm="forward"), type=2 - self.lobatto, n=N, axis=-1, ) * (N - self.lobatto) - return fq def harmonics(self): """Spectral coefficients aₘₙ of the interpolating trigonometric polynomial. diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 679c2f9d1..04f144af7 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -1,10 +1,9 @@ """Methods for computing bounce integrals (singular or otherwise).""" from interpax import CubicHermiteSpline, PPoly -from numpy.fft import rfft2 from orthax.legendre import leggauss -from desc.backend import dct, jnp +from desc.backend import dct, jnp, rfft2 from desc.integrals.basis import FourierChebyshevSeries, PiecewiseChebyshevSeries from desc.integrals.bounce_utils import ( _bounce_quadrature, @@ -396,6 +395,7 @@ def __init__( self, grid, data, + iota, theta, N_B=None, num_transit=16, @@ -430,6 +430,9 @@ def __init__( data : dict[str, jnp.ndarray] Data evaluated on ``grid``. Must include names in ``Bounce2D.required_names``. + iota : jnp.ndarray + Shape (L, ). + Rotational transform. theta : jnp.ndarray Shape (L, M, N). DESC coordinates θ sourced from the Clebsch coordinates @@ -488,18 +491,7 @@ def __init__( self._x, self._w = get_quadrature(quad, automorphism) # peel off field lines - iota = data["iota"].ravel() - alpha = get_alpha( - alpha, - iota=( - grid.compress(iota) - if iota.size == grid.num_nodes - # assume passed in reshaped data over flux surface - else jnp.array(iota[0]) - ), - num_transit=num_transit, - period=2 * jnp.pi, - ) + alpha = get_alpha(alpha, iota=iota, num_transit=num_transit, period=2 * jnp.pi) # Compute spectral coefficients. self._T, self._B = _transform_to_clebsch_1d( grid, alpha, theta, data["|B|"] / Bref, N_B, is_reshaped @@ -513,7 +505,7 @@ def __init__( @staticmethod def compute_theta(eq, M=16, N=32, rho=1.0, clebsch=None, **kwargs): - """Return DESC coordinates θ of Fourier Chebyshev basis nodes. + """Return DESC coordinates θ of (α,ζ) Fourier Chebyshev basis nodes. Parameters ---------- @@ -1091,10 +1083,10 @@ def __init__( self._x, self._w = get_quadrature(quad, automorphism) # Compute local splines. - self._zeta = grid.compress(grid.nodes[:, 2], surface_label="zeta") + self.zeta = grid.compress(grid.nodes[:, 2], surface_label="zeta") self.B = jnp.moveaxis( CubicHermiteSpline( - x=self._zeta, + x=self.zeta, y=self._data["|B|"], dydx=self._data["|B|_z|r,a"], axis=-1, @@ -1103,7 +1095,7 @@ def __init__( source=(0, 1), destination=(-1, -2), ) - self._dB_dz = polyder_vec(self.B) + self.dB_dz = polyder_vec(self.B) # Add axis here instead of in ``_bounce_quadrature``. for name in self._data: @@ -1164,7 +1156,7 @@ def points(self, pitch_inv, *, num_well=None): line and pitch, is padded with zero. """ - return bounce_points(pitch_inv, self._zeta, self.B, self._dB_dz, num_well) + return bounce_points(pitch_inv, self.zeta, self.B, self.dB_dz, num_well) def check_points(self, points, pitch_inv, *, plot=True, **kwargs): """Check that bounce points are computed correctly. @@ -1197,7 +1189,7 @@ def check_points(self, points, pitch_inv, *, plot=True, **kwargs): z1=points[0], z2=points[1], pitch_inv=pitch_inv, - knots=self._zeta, + knots=self.zeta, B=self.B, plot=plot, **kwargs, @@ -1215,6 +1207,7 @@ def integrate( batch=True, check=False, plot=False, + quad=None, ): """Bounce integrate ∫ f(λ, ℓ) dℓ. @@ -1262,6 +1255,9 @@ def integrate( plot : bool Whether to plot the quantities in the integrand interpolated to the quadrature points of each integral. Ignored if ``check`` is false. + quad : tuple[jnp.ndarray] + Optional quadrature points and weights. If given this overrides + the quadrature chosen when this object was made. Returns ------- @@ -1274,14 +1270,14 @@ def integrate( if points is None: points = self.points(pitch_inv) result = _bounce_quadrature( - x=self._x, - w=self._w, + x=self._x if quad is None else quad[0], + w=self._w if quad is None else quad[1], integrand=integrand, points=points, pitch_inv=pitch_inv, f=setdefault(f, []), data=self._data, - knots=self._zeta, + knots=self.zeta, method=method, batch=batch, check=check, @@ -1291,9 +1287,9 @@ def integrate( result *= interp_to_argmin( weight, points, - self._zeta, + self.zeta, self.B, - self._dB_dz, + self.dB_dz, method, ) return result @@ -1319,7 +1315,7 @@ def plot(self, m, l, pitch_inv=None, **kwargs): Matplotlib (fig, ax) tuple. """ - B, dB_dz = self.B, self._dB_dz + B, dB_dz = self.B, self.dB_dz if B.ndim == 4: B = B[m] dB_dz = dB_dz[m] @@ -1331,9 +1327,9 @@ def plot(self, m, l, pitch_inv=None, **kwargs): pitch_inv.ndim > 1, msg=f"Got pitch_inv.ndim={pitch_inv.ndim}, but expected 1.", ) - z1, z2 = bounce_points(pitch_inv, self._zeta, B, dB_dz) + z1, z2 = bounce_points(pitch_inv, self.zeta, B, dB_dz) kwargs["z1"] = z1 kwargs["z2"] = z2 kwargs["k"] = pitch_inv - fig, ax = plot_ppoly(PPoly(B.T, self._zeta), **_set_default_plot_kwargs(kwargs)) + fig, ax = plot_ppoly(PPoly(B.T, self.zeta), **_set_default_plot_kwargs(kwargs)) return fig, ax diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index d85220a4f..1f4ea2049 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -51,7 +51,8 @@ def get_alpha(alpha_0, iota, num_transit, period): """ # Δϕ (∂α/∂ϕ) = Δϕ ι̅ = Δϕ ι/2π = Δϕ data["iota"] - alpha = alpha_0 + period * iota[:, jnp.newaxis] * jnp.arange(num_transit) + # FIXME: Looks innocent, but JAX doesn't like this... + alpha = alpha_0 + period * jnp.expand_dims(iota, -1) * jnp.arange(num_transit) return alpha diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 9de919126..86cb5b575 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -1230,7 +1230,7 @@ def dg_dz(z): points = (np.array(0, ndmin=4), np.array(2 * np.pi, ndmin=4)) argmin = 5.61719 h_min = h(argmin) - result = func(h(zeta), points, zeta, bounce.B, bounce._dB_dz) + result = func(h(zeta), points, zeta, bounce.B, bounce.dB_dz) assert result.shape == points[0].shape np.testing.assert_allclose(h_min, result, rtol=1e-3) @@ -1578,7 +1578,13 @@ def test_bounce2d_checks(self): theta = Bounce2D.compute_theta(eq, M=8, N=64, rho=rho) # 5. Make the bounce integration operator. bounce = Bounce2D( - grid, data, theta, num_transit=2, quad=leggauss(3), check=True + grid, + data, + iota=grid.compress(data["iota"]), + theta=theta, + num_transit=2, + quad=leggauss(3), + check=True, ) pitch_inv, _ = bounce.get_pitch_inv_quad( min_B=grid.compress(data["min_tz |B|"]), @@ -1686,6 +1692,7 @@ def test_binormal_drift_bounce2d(self): bounce = Bounce2D( grid=grid, data=grid_data, + iota=data["iota"], theta=Bounce2D.compute_theta( eq, M, From ce64c53b7f6dfbde5b4f8e5e28e0e9b95f28230f Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 02:08:19 -0400 Subject: [PATCH 39/44] Add missing key to stability fun test --- tests/test_stability_funs.py | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/tests/test_stability_funs.py b/tests/test_stability_funs.py index 72b4819b3..7233b5ce4 100644 --- a/tests/test_stability_funs.py +++ b/tests/test_stability_funs.py @@ -416,6 +416,7 @@ def test_ballooning_geometry(tmpdir_factory): "g^ra", "g^rr", "cvdrift", + "gbdrift", "cvdrift0", "|B|", "B^zeta", @@ -448,11 +449,9 @@ def test_ballooning_geometry(tmpdir_factory): sign_psi = psi_s / np.abs(psi_s) sign_iota = iotas / np.abs(iotas) - modB = data["|B|"] x = Lref * np.sqrt(psi) shat = -x / iotas * shears / Lref - psi_r = data["psi_r"] grad_alpha = data["grad(alpha)"] g_sup_rr = data["g^rr"] From 7c882115d526f60dc9110c5312a04a9e91d527f8 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 16:38:46 -0400 Subject: [PATCH 40/44] Document algorithm choices --- desc/integrals/bounce_integral.py | 34 ++++++++++++++++++++----------- desc/integrals/bounce_utils.py | 7 ++++++- desc/integrals/interp_utils.py | 23 +++++++++++++-------- 3 files changed, 42 insertions(+), 22 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 04f144af7..583c75f11 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -289,9 +289,8 @@ class Bounce2D(IOAble): polynomials are preferred to other orthogonal polynomial series since fast discrete polynomial transforms (DPT) are implemented via fast transform to Chebyshev then DCT. Although nothing prohibits a direct DPT, we want to - rely on existing libraries. There are other reasons to prefer Chebyshev series - not discussed here. Therefore, a Fourier-Chebyshev series is chosen to - interpolate θ(α,ζ), and a piecewise Chebyshev series interpolates |B|(ζ). + rely on existing libraries. Therefore, a Fourier-Chebyshev series is chosen + to interpolate θ(α,ζ), and a piecewise Chebyshev series interpolates |B|(ζ). Computing accurate series expansions in (α, ζ) coordinates demands particular interpolation points in that coordinate system. Newton iteration @@ -318,9 +317,12 @@ class Bounce2D(IOAble): The reason θ is not interpolated with a double Fourier series θ(ϑ, ζ) is because quadrature points along |B|(α=α₀, ζ) can be identified by a single variable; evaluating the multivariable map θ(ϑ(α, ζ), ζ) is expensive - compared to evaluating the single variable map θ(α=α₀, ζ). - Another option is to use a filtered Fourier series, - doi.org/10.1016/j.aml.2006.10.001. + compared to evaluating the single variable map θ(α=α₀, ζ). Also, the advantage + of DESC coordinates is that they use the spectrally condensed variable + ζ* = NFP ζ. This cannot be done in any other coordinate system, regardless of + whether it is periodic or not, so (ϑ, ϕ) coordinates are no better than (α, ζ) + coordinates in this aspect. (Another option is to use a filtered Fourier + series, doi.org/10.1016/j.aml.2006.10.001). After computing the bounce points, the supplied quadrature is performed. By default, this is a Gauss quadrature after removing the singularity. @@ -332,10 +334,8 @@ class Bounce2D(IOAble): efficiently, and the fixed nature of quadrature performs better on GPUs. Fast transforms are used where possible. Fast multipoint methods are not - implemented. For non-uniform interpolation, MMTs are used. It should be - worthwhile to use the inverse non-uniform fast transforms, so long as the - quadrature packs nodes at reasonable density. Fast multipoint methods are - preferable because they are exact, but that requires more development work. + implemented. For non-uniform interpolation, MMTs are used. It will be + worthwhile to use the inverse non-uniform fast transforms. Additional notes on multivalued coordinates. The definition of α in B = ∇ρ × ∇α on an irrational magnetic surface @@ -829,10 +829,20 @@ def compute_length(self, quad=None): Shape (L, ). """ - # Gauss quadrature captures double frequency of Chebyshev series. + # Integrating an analytic map, so a fixed high order quadrature is ideal. + # The integration domain is not periodic, so the best candidates to choose + # are Gauss-Legendre quadrature and Clenshaw-Curtis. GL is more efficient + # than CC for analytic maps by a factor of 2. Advantage of CC is that θ at + # the quadrature points can be computed with fast cosine transform. However, + # one still needs to perform a non-uniform inverse fourier transform of B^ζ + # at those points, which will be the dominating expense. The spectral width + # of θ along field lines is also narrower than B^ζ, especially at high NFP. x, w = leggauss(self._B.N // 2) if quad is None else quad - # TODO: There exits a fast transform from Chebyshev series to Legendre nodes. + # TODO: Use fast Chebyshev to Legendre inverse transform. + # When converted to a Legendre series, θ at the quadrature points can + # be computed with a fast transform (without needing to compute leggauss nodes). + # This is likely preferable to even a true non-uniform transform. theta = idct_non_uniform(x, self._T.cheb[..., jnp.newaxis, :], self._T.N) zeta = jnp.broadcast_to(bijection_from_disc(x, 0, 2 * jnp.pi), theta.shape) diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 1f4ea2049..511747daf 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -51,7 +51,12 @@ def get_alpha(alpha_0, iota, num_transit, period): """ # Δϕ (∂α/∂ϕ) = Δϕ ι̅ = Δϕ ι/2π = Δϕ data["iota"] - # FIXME: Looks innocent, but JAX doesn't like this... + # FIXME: Looks innocent, but JAX doesn't like this. It works fine if iota + # is already an array of ndim 1, but when we loop over rho in the registered + # compute fun to calculate epsilon effective, iota becomes a float (jax complains + # it's an int64[] for some reason) and when it is a scalar instead of an array + # jax thinks the computation needs to know the particular value of iota to jit + # beforehand. when it is an array jax should treat it without issue as usual. alpha = alpha_0 + period * jnp.expand_dims(iota, -1) * jnp.arange(num_transit) return alpha diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index b600c6f84..a837273bb 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -24,7 +24,7 @@ # TODO: Transformation to make nodes more uniform Boyd eq. 16.46 pg. 336. -# More uniformly spaced nodes will speed up convergence. +# More uniformly spaced nodes might speed up convergence. def cheb_pts(N, domain=(-1, 1), lobatto=False): @@ -141,15 +141,20 @@ def harmonic_vander(x, M, domain=(0, 2 * np.pi)): return basis -# TODO: For inverse transforms, do multipoint evaluation with FFT. -# FFT cost is 𝒪(M N log[M N]) while direct evaluation is 𝒪(M² N²). -# Chapter 10, https://doi.org/10.1017/CBO9781139856065. -# Right now we do an MMT with the Vandermode matrix. -# Multipoint is likely better than using NFFT (for strong singular bounce -# integrals) to evaluate f(xq) given fourier coefficients because evaluation -# points are quadratically packed near edges for efficient quadrature. For -# weak singularities (e.g. effective ripple) NFFT should work well. +# TODO: For inverse transforms, use non-uniform fast transforms (NFFT). # https://github.com/flatironinstitute/jax-finufft. +# Let spectral resolution be F, (e.g. F = M N for 2D transform), +# and number of points (non-uniform) to evaluate be Q. A non-uniform +# fast transform cost is 𝒪([F+Q] log[F] log[1/ε]) where ε is the +# interpolation error term (depending on implementation how ε appears +# may change, but it is always logarithmic). Direct evaluation is 𝒪(F Q). +# Note that for the inverse Chebyshev transforms, we can also use fast +# multipoint methods Chapter 10, https://doi.org/10.1017/CBO9781139856065. +# Unlike NFFTs, multipoint methods are exact and reduce to using FFTs. +# The cost is 𝒪([F+Q] łog²[F + Q]). This is a good candidate for evaluating +# |B|, since the integrands are not smooth functions of |B|, which we know +# as a Chebyshev series, and the nodes are packed more tightly near the edges, +# in particular for the strongly singular integrals. def interp_rfft(xq, f, domain=(0, 2 * jnp.pi), axis=-1): From 1376e906c4469299f3f37c59f8f4196a1a0de0b4 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Sun, 6 Oct 2024 16:54:24 -0400 Subject: [PATCH 41/44] Update incorrect comment --- desc/integrals/bounce_integral.py | 10 ++++------ 1 file changed, 4 insertions(+), 6 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 583c75f11..ac002b1c9 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -143,10 +143,10 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Now, it appears the Fourier transform of θ may have small oscillatory bumps outside reasonable bandwidths. This impedes full convergence of any approximation, and in particular the poloidal Fourier series for, θ(α, ζ=ζ₀). - Maybe this is because θ is not a physical signal as it is determined by the - optimizer. (Note the Fourier series converges fast for |B|, even in - non-omnigenous configurations where (∂|B|/∂α)|ρ,ζ is not small, so this is - indeed some feature with θ). + Maybe this is because the Chebyshev interpolation is detected root-finding + errors where the nodes are more densely clustered θ. (Note the Fourier series + converges fast for |B|, even in non-omnigenous configurations where + (∂|B|/∂α)|ρ,ζ is not small, so this is indeed some feature with θ). Therefore, we explicitly enforce continuity of our approximation of θ between cuts to short-circuit the convergence. This works to remove the small @@ -418,8 +418,6 @@ def __init__( ----- Performance may improve significantly if the spectral resolutions ``m``, ``n``, ``M``, ``N``, and ``N_B`` are powers of two. - It is possible for the optimal ``M``, ``N``, and ``N_B`` to decrease - when ``m`` and ``n`` increase, though ``M`` is usually still large. Parameters ---------- From d54a80991845417d5fbe3076595a3ca31f1867ee Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Mon, 7 Oct 2024 02:25:11 -0400 Subject: [PATCH 42/44] document better root finding method for later --- desc/integrals/bounce_integral.py | 6 +++--- desc/integrals/interp_utils.py | 5 +++++ 2 files changed, 8 insertions(+), 3 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index ac002b1c9..70cdcf70e 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -143,8 +143,8 @@ def _transform_to_clebsch_1d(grid, alpha, theta, B, N_B, is_reshaped=False): Now, it appears the Fourier transform of θ may have small oscillatory bumps outside reasonable bandwidths. This impedes full convergence of any approximation, and in particular the poloidal Fourier series for, θ(α, ζ=ζ₀). - Maybe this is because the Chebyshev interpolation is detected root-finding - errors where the nodes are more densely clustered θ. (Note the Fourier series + Maybe this is because the Chebyshev interpolation is detecting root-finding + errors where the nodes are more densely clustered. (Note the Fourier series converges fast for |B|, even in non-omnigenous configurations where (∂|B|/∂α)|ρ,ζ is not small, so this is indeed some feature with θ). @@ -311,7 +311,7 @@ class Bounce2D(IOAble): Recall that periodicity enables faster convergence, motivating the desire to instead interpolate |B|(ϑ, ϕ) with a double Fourier series and applying - bisection methods to find bounce points with mesh size inversely + bisection methods to find roots with mesh size inversely proportional to the max frequency along the field line: M ι + N. ``Bounce2D`` does not use that approach as that root-finding scheme is inferior. The reason θ is not interpolated with a double Fourier series θ(ϑ, ζ) is diff --git a/desc/integrals/interp_utils.py b/desc/integrals/interp_utils.py index a837273bb..cfff90487 100644 --- a/desc/integrals/interp_utils.py +++ b/desc/integrals/interp_utils.py @@ -20,6 +20,11 @@ # TODO: Boyd's method 𝒪(N²) instead of Chebyshev companion matrix 𝒪(N³). # John P. Boyd, Computing real roots of a polynomial in Chebyshev series # form through subdivision. https://doi.org/10.1016/j.apnum.2005.09.007. +# Use that once to find extrema of |B| if N_B > 64. Then to find roots +# of bounce points use the closed formula in Boyd's spectral methods +# section 19.6. Can isolate interval to search for root by observing +# whether B - 1/pitch changes sign at extrema. This is significantly +# cheaper and non-iterative, so jax and gpu will like it. chebroots_vec = jnp.vectorize(chebroots, signature="(m)->(n)") From 8cef6d39e34d456f7061a9aa5178a0a33dc83810 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Mon, 7 Oct 2024 12:46:30 -0400 Subject: [PATCH 43/44] update ballooning notebook by explicitly comping gbdrift --- docs/notebooks/tutorials/ideal_ballooning_stability.ipynb | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/notebooks/tutorials/ideal_ballooning_stability.ipynb b/docs/notebooks/tutorials/ideal_ballooning_stability.ipynb index 1ccb88234..a2ffbf30b 100644 --- a/docs/notebooks/tutorials/ideal_ballooning_stability.ipynb +++ b/docs/notebooks/tutorials/ideal_ballooning_stability.ipynb @@ -783,6 +783,7 @@ } ], "source": [ + "ball_data0 = eq1.compute(\"gbdrift\", grid=grid, data=ball_data0)\n", "fig, ax = plt.subplots(7, sharex=True, figsize=(8, 16))\n", "\n", "ax[0].plot(zeta, (ball_data0[\"B^zeta\"] / ball_data0[\"|B|\"])[::nalpha], \"-or\", ms=1.5)\n", @@ -827,7 +828,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.9" + "version": "3.12.6" } }, "nbformat": 4, From 0e07215ff42f3f3fc3a3c3a55eb20c8da0786981 Mon Sep 17 00:00:00 2001 From: unalmis <kayaunalmis@proton.me> Date: Mon, 7 Oct 2024 16:20:31 -0400 Subject: [PATCH 44/44] resolving jit error --- desc/integrals/bounce_integral.py | 2 +- desc/integrals/bounce_utils.py | 6 ------ tests/test_axis_limits.py | 4 ++-- tests/test_integrals.py | 15 ++++++++------- 4 files changed, 11 insertions(+), 16 deletions(-) diff --git a/desc/integrals/bounce_integral.py b/desc/integrals/bounce_integral.py index 70cdcf70e..aea90fe00 100644 --- a/desc/integrals/bounce_integral.py +++ b/desc/integrals/bounce_integral.py @@ -489,7 +489,7 @@ def __init__( self._x, self._w = get_quadrature(quad, automorphism) # peel off field lines - alpha = get_alpha(alpha, iota=iota, num_transit=num_transit, period=2 * jnp.pi) + alpha = get_alpha(alpha, iota, num_transit, 2 * jnp.pi) # Compute spectral coefficients. self._T, self._B = _transform_to_clebsch_1d( grid, alpha, theta, data["|B|"] / Bref, N_B, is_reshaped diff --git a/desc/integrals/bounce_utils.py b/desc/integrals/bounce_utils.py index 511747daf..26e2756c4 100644 --- a/desc/integrals/bounce_utils.py +++ b/desc/integrals/bounce_utils.py @@ -51,12 +51,6 @@ def get_alpha(alpha_0, iota, num_transit, period): """ # Δϕ (∂α/∂ϕ) = Δϕ ι̅ = Δϕ ι/2π = Δϕ data["iota"] - # FIXME: Looks innocent, but JAX doesn't like this. It works fine if iota - # is already an array of ndim 1, but when we loop over rho in the registered - # compute fun to calculate epsilon effective, iota becomes a float (jax complains - # it's an int64[] for some reason) and when it is a scalar instead of an array - # jax thinks the computation needs to know the particular value of iota to jit - # beforehand. when it is an array jax should treat it without issue as usual. alpha = alpha_0 + period * jnp.expand_dims(iota, -1) * jnp.arange(num_transit) return alpha diff --git a/tests/test_axis_limits.py b/tests/test_axis_limits.py index b8d173f7e..a2f556264 100644 --- a/tests/test_axis_limits.py +++ b/tests/test_axis_limits.py @@ -61,12 +61,12 @@ "g^tz_r", "g^tz_t", "g^tz_z", - "grad(alpha)", "g^aa", "g^ra", + "grad(alpha)", + "periodic(grad(alpha))", "gbdrift", "cvdrift", - "grad(alpha)", "|e^helical|", "|grad(theta)|", "<J*B> Redl", # may not exist for all configurations diff --git a/tests/test_integrals.py b/tests/test_integrals.py index 86cb5b575..90cc80485 100644 --- a/tests/test_integrals.py +++ b/tests/test_integrals.py @@ -13,7 +13,7 @@ from scipy.special import ellipe, ellipkm1 from tests.test_plotting import tol_1d -from desc.backend import jnp +from desc.backend import jit, jnp from desc.basis import FourierZernikeBasis from desc.equilibrium import Equilibrium from desc.equilibrium.coords import get_rtz_grid @@ -1522,7 +1522,7 @@ def fun2(pitch): class TestBounce2D: - """Test bounce integration with two-dimensional pseudo-spectral methods.""" + """Test bounce integration that uses 2D pseudo-spectral methods.""" @pytest.mark.unit @pytest.mark.parametrize( @@ -1533,11 +1533,12 @@ class TestBounce2D: ], ) def test_alpha_sequence(self, alpha_0, iota, num_period, period): - """Test field line label is updated correctly after toroidal transits.""" - # this test will be more useful once zeta != phi - iota = np.atleast_1d(iota) - alphas = get_alpha(alpha_0, iota, num_period, period) - assert alphas.shape == (iota.size, num_period) + """Test field line label updating works with jit.""" + alphas = jit(get_alpha, static_argnums=2)(alpha_0, iota, num_period, period) + if np.ndim(iota): + assert alphas.shape == (iota.size, num_period) + else: + assert alphas.shape == (num_period,) print(alphas) @staticmethod

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