numerical-methods-montecarlointegration-exercise

In this exercise, you should implement classes that provide a reasonably flexible framework for Monte-Carlo integration in arbitrary dimensions.

The framework should be flexible enough to allow

• integration of different functions f : Rⁿ → R

• flexible specification of the integration domains using a transformation from [0,1]ⁿ to a subset of Rⁿ

• flexible use for different random number generators

For convenience, we provide the interfaces that define the framework. You can find these interfaces in the package

info.quantlab.numericalmethods.lecture.montecarlo.integration

of the project numerical-methods-lecture, see github.com/qntlb/numerical-methods-lecture. This project is already defined as a Maven dependency to this project. This project is pre-configured and "knows" these interfaces.

Interfaces (provided)

• Integrand
• IntegrationDomain
• Integrator
• MonteCarloIntegratorFactory

See the package info.quantlab.numericalmethods.lecture.montecarlo.integration

The MonteCarloIntegratorFactory's method requires a class implementing a RandomNumberGenerator. This interface and some classes implementing this interface can be found in the package

info.quantlab.numericalmethods.lecture.randomnumbers

Integrand and IntegrationDomain

• Objects implementing Integrand provide a function f : A → R defined on a domain A ⊂ Rⁿ.
• Objects implementing IntegrationDomain provide a bijective function g : [0,1]ⁿ → A that transforms the integration domain and the determinant of the derivative (Jacobi matrix) dg/dx.
• Objects implementing Integrator provide the integral ∫_A f(z) dz using substitution z = f(x).

Classes

You may use the classes providing random number generators that will be or were developed during the lecture, e.g.,

• RandomNumberGeneratorFrom1D
• MersenneTwister

Task

The exercise consists of two separate tasks.

Task 1: A MonteCarloIntegrator

To complete your task:

• 1. Implement a class implementing the interface Integrator that performs a general Monte-Carlo integration of arbitrary functions on general domains.

The function to integrate will be provided to the integrator's method integrate as an object implementing the interface Integrand.

The integration domain will be provided to the integrator's method integrate as an object implementing the interface IntegrationDomain.

• 2. Implement a class implementing the interface MonteCarloIntegratorFactory that allows creating an object of the class that you have implemented in 1). Note: the MonteCarloIntegratorFactory simply calls the constructor of your class.
• 3. To allow us to test you implementation, complete the implementation of the method getMonteCarloIntegratorFactory of MonteCarloIntegrationAssignmentSolution. This allows the creation of an object of your MonteCarloIntegratorFactory. Our unit tests will use this to test your code.
• 4. Feel free to create your own UnitTests and JavaDoc documentation.

Suggestion: you may test your integrator with different random number generators, e.g. MersenneTwister via

	final long seed = 3141;
	RandomNumberGenerator randomNumberGenerator = new RandomNumberGeneratorFrom1D(new MersenneTwister(seed), domain.getDimension());

or a HaltonSequence.

Note: As known from the lecture, using a 1-dimensional quasi-random number sequence in a d-dimensional integration will lead to wrong results. Hence, it makes usage of your integrator safer, if you explcitly require that the dimension of the domain matches the dimension of the random number sequence (an throw IllegalArgumentException if not), such that the user has to explicitly use RandomNumberGeneratorFrom1D when appropriate. You could however allow that an n-dimension sequence is used for a d-dimensional domain if n>d.

Task 2: Using your MonteCarloIntegrator to calculate the integral of a DoubleBinaryFunction

• 5. Complete the method getIntegral of MonteCarloIntegrationAssignmentSolution. Use your Monte-Carlo integrator with approximately 1 million sample points to calculate the integral.

Tasks 3: Implement a SimpsonsIntegrator for the general Simpson's rule in d dimension

To complete your task:

• 6. Implement a class implementing the interface Integrator that performs a general (composite) Simpson's rule integration in d dimension of arbitrary functions on general domains.

The function to integrate will be provided to the integrator's method integrate as an object implementing the interface Integrand.

The integration domain will be provided to the integrator's method integrate as an object implementing the interface IntegrationDomain.

• 7. Implement a class implementing the interface IntegratorFactory that allows creating an object of the class that you have implemented in 1). Note: the IntegratorFactory simply calls the constructor of your class.
• 8. To allow us to test you implementation, complete the implementation of the method getSimpsonsIntegratorFactory of MonteCarloIntegrationAssignment. This allows the creation of an object of your IntegratorFactory. Our unit tests will use this to test your code.
• 9. Feel free to create your own UnitTests and JavaDoc documentation.

Hints

• Note that your Simpson's integral and your Monte-Carlo integral only operator on [0,1]^d (the object implementing the Domain will provide you with the transformation).

• Your Simpson's integrator should accept the numberOfValuationPoints as an argument. This should be the minimum total number of valuation points. Since the Simpsons rule uses an odd number of points in every dimension, you may use the following code to round this number appropriately to numberOfSamplePointsEffective, using numberOfSamplePointsPerDimension per dimension.

	int dimension = integrationDomain.getDimension();
	int numberOfValuationPointsPerDimension = 2 * (int) (Math.ceil(Math.pow(numberOfValuationPoints, 1.0/dimension))/2) + 1;
	int numberOfValuationPointsEffective = (int) Math.pow(numberOfValuationPointsPerDimension, dimension);

• You might realise that you need to think a bit to find a short algorithm to implement the Simpsons integration in arbitrary dimensions. It is possible to create a fairly short implementation if you implement a multi-index index - an array of length dimension where each entry runs from 0 to numberOfSamplePointsPerDimension-1.

Task 4 (optional): Use your MonteCarloIntegrator with an RandomNumberGenerator generating a low-discrepancy in d dimensions.

Since the random number generator is an input to your implementation of the MonteCarloIntegrator interface, it should be possible to feed in low discrepancy sequences (given that they implement the RandomNumberGeneratorRandomNumberGenerator interface).

Conduct a an experiment using low discrepancy sequence to perform the Monte-Carlo integration and compare the integration error to that of using a pseudo random number sequence.

Unit Tests

We encourage you to write your own unit tests.

Further Research

This project offers the opportunity to explore Monte-Carlo integration in more detail for those interested. Here are a few suggestions:

• Explore the dependency on the dimension: Consider the integration of x → product(i=0,...,d-1) sin(x_i) for 0 < x_i < π. The value of the integral is 2^d. This is an d-dimensional integral. For this function, compare the accuracy of Monte-Carlo integration and Simpsons integration with d = 1, 2, 4, 8 using for example n = 5^8 = 390625 sample points.

• Explore the dependency on the smoothness of the function: Consider the integration of (x₀,x₁) → x₀² + x₁² < 1.0 ? 1.0 : 0.0 for 0 < x_i < 1. The analytic value of this integral π. For this function, compare the accuracy of Monte-Carlo integration and Simpsons integration using n = 101^2 = 10201 sample points.