Nonlinear Preconditioner Support

[termi-official]

Is your feature request related to a problem? Please describe.

Solving nonlinear problems to find $u$ such that $F(u) = 0$ can be made possible and/or be speed up by providing left and right nonlinear preconditioners, i.e. some other functions $H$ and $G$, such that $G(F(H(\tilde{u}))) = 0$ and $u = H(\tilde{u})$, where $G$, $H$ and $F$ share the same roots. I think [1] gives a nice start here.

I want to solve two problems here.

1. Support for actual preconditioning as e.g. in ASPIN or Walkers-Xi Anderson mixing.
2. Support for Dirichlet constraints (see What is a Nonlinear Solver? And How to easily build Newer Ones #345 (comment) for some discussion), where $H$ enforces the constraint to the solution and $G \circ F$ is the condensed problem.

Describe the solution you’d like

I think we can come up with a similar solution as in LinearSolve.jl, to pass $H$ and $G$ into the constructor and applying them at appropriate positions in the respective algorithms.

Describe alternatives you’ve considered

The only alternative which I see for now is, that we can define custom solvers (e.g. ASPIN) directly. However, I think the idea above leads to better composability.

Additional context

See SciML/OrdinaryDiffEq.jl#1570 for more previous discussion related to this topic.

References

[1] Brune, Peter R., et al. "Composing scalable nonlinear algebraic solvers." SIAM Review 57.4 (2015): 535-565.

[colegruninger97]

Hello, I am curious about the status of this issue. I am also interested in composing nonlinear solvers for the purposes of preconditioning implicit formulations of stiff, nonlinear pde. It would be great to be able to do this in Julia using NonlinearSolve.jl for fast prototyping. Are there any updates or planned developments of such a feature? Thanks.

[termi-official]

At least from my end I am nowhere close to resolving this anytime soon unfortunately.