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SciML / LinearSolve.jl

LinearSolve.jl: High-Performance Unified Interface for Linear Solvers in Julia. Easily switch between factorization and Krylov methods, add preconditioners, and all in one interface.
https://docs.sciml.ai/LinearSolve/stable/
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Using preconditioners in the adjoints of a linear solve #476

Open ChrisRackauckas opened 4 months ago

ChrisRackauckas commented 4 months ago

From @avik-pal https://github.com/SciML/LinearSolve.jl/pull/449

Preconditioning. These need to be added to the Adjoint Sensitivity Struct. Can we use a left preconditioner for the forward problem as the transpose right preconditioner? Need someone with a better grasp of linear algebra to chime in here

@amontoison I think some Krylov methods only support pre-conditioners on one side though?

Maybe we need some way for someone to specify adjoint preconditioners?

amontoison commented 4 months ago

Hi @ChrisRackauckas, Krylov methods dedicated to symmetric linear systems only support centered preconditioning (special case of left + right preconditioning) to maintain the symmetry. But it should not be an issue for the adjoint sensitivity because A = A', so you can reuse the same preconditioner.

If the system is not symmetric, the Krylov method supports both left and right preconditioning.

For the transpose of the preconditioner, I expect the user to provide adjoint(P) as input of the solver. But it's not always easy to do that when P is a matrix-free preconditioner.