Cheap inverse of A+kI if cheap inverse from A

[gaurav-arya]

If A has a cheap inverse, A+kI nearly always does too. It is very important to exploit this, e.g. for cheap inverses of the W-operator in implicit ODE solves: if the user provides a Jacobian with a cheap inverse, we don't want to waste it.

The solution isn't super clear, but some ideas:

• MatrixOperator(A) + k*I = MatrixOperator(A + k*I), rather than AddedOperator(...)
• Similarly, additions to identity should be specialized upon for other types, e.g. tensor product operators. The idea is to avoid a lazy addition when possible to avoid the need for Krylov solves.
• When a user specifies F = FunctionOperator(...), somehow we should make it that F+kI is also a FunctionOperator when possible, e.g. maybe the user specifies a general class of ops that work for arbitrary k. (This one is vague, and definitely needs design work, but hopefully we can find an elegant solution.)
• As a last resort, we can also consider things like the Sherman-Morrison formula which apply generically.

[vpuri3]

what's expression for the inverse of (A + kI)?

[ChrisRackauckas]

It's moreso that sometimes it's structured. For example you can know that you keep positive-definite.

[gaurav-arya]

Regarding Woodbury, see bottom of page 3 here: https://arxiv.org/pdf/2309.03060.pdf