[petsc-users] using preconditioner with SLEPc

[Jose E. Roman]

The problem can be written as A0*v=omega*B0*v and you want the eigenvalues omega closest to zero. If the matrices were explicitly available, you would do shift-and-invert with target=0, that is

  (A0-sigma*B0)^{-1}*B0*v=theta*v    for sigma=0, that is

  A0^{-1}*B0*v=theta*v

and you compute EPS_LARGEST_MAGNITUDE eigenvalues theta=1/omega.

Matt: I guess you should have EPS_LARGEST_MAGNITUDE instead of EPS_SMALLEST_REAL in your code. Are you getting the eigenvalues you need? EPS_SMALLEST_REAL will give slow convergence.

Florian: I would not recommend setting the KSP matrices directly, it may produce strange side-effects. We should have an interface function to pass this matrix. Currently there is STPrecondSetMatForPC() but it has two problems: (1) it is intended for STPRECOND, so cannot be used with Krylov-Schur, and (2) it is not currently available in the python interface.

The approach used by Matt is a workaround that does not use ST, so you can handle linear solves with a KSP of your own.

As an alternative, since your problem is symmetric, you could try LOBPCG, assuming that the leftmost eigenvalues are those that you want (e.g. if all eigenvalues are non-negative). In that case you could use STPrecondSetMatForPC(), but the remaining issue is calling it from python.

If you are using the git repo, I could add the relevant code.

Jose



> El 8 feb 2021, a las 14:22, Matthew Knepley <knepley at gmail.com> escribió:
> 
> On Mon, Feb 8, 2021 at 7:04 AM Florian Bruckner <e0425375 at gmail.com> wrote:
> Dear PETSc / SLEPc Users,
> 
> my question is very similar to the one posted here: 
> https://lists.mcs.anl.gov/pipermail/petsc-users/2018-August/035878.html
> 
> The eigensystem I would like to solve looks like:
> B0 v = 1/omega A0 v
> B0 and A0 are both hermitian, A0 is positive definite, but only given as a linear operator (matshell). I am looking for the largest eigenvalues (=smallest omega). 
> 
> I also have a sparse approximation P0 of the A0 operator, which i would like to use as precondtioner, using something like this:
> 
>         es = SLEPc.EPS().create(comm=fd.COMM_WORLD)
>         st = es.getST()
>         ksp = st.getKSP()
>         ksp.setOperators(self.A0, self.P0)
> 
> Unfortunately PETSc still complains that it cannot create a preconditioner for a type 'python' matrix although P0.type == 'seqaij' (but A0.type == 'python'). 
> By the way, should P0 be an approximation of A0 or does it have to include B0?
> 
> Right now I am using the krylov-schur method. Are there any alternatives if A0 is only given as an operator?
> 
> Jose can correct me if I say something wrong.
> 
> When I did this, I made a shell operator for the action of A0^{-1} B0 which has a KSPSolve() in it, so you can use your P0 preconditioning matrix, and
> then handed that to EPS. You can see me do it here:
> 
>   https://gitlab.com/knepley/bamg/-/blob/master/src/coarse/bamgCoarseSpace.c#L123
> 
> I had a hard time getting the embedded solver to work the way I wanted, but maybe that is the better way.
> 
>   Thanks,
> 
>      Matt
>  
> thanks for any advice
> best wishes
> Florian
> 
> 
> -- 
> What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.
> -- Norbert Wiener
> 
> https://www.cse.buffalo.edu/~knepley/