[petsc-users] How to compute the condition number of SchurComplementMat preconditioned with PCSHELL.

[Vladislav Pimanov]

Dear Barry,


Let S = B A^{-1} B^T be the Schur complement, and \hat{S} = B diag(A)^{-1} B^T denotes the preconditioner.

I also tried KSPComputeExtremeSingularValues for rtol(A) = rtol(\hat{S}) = 1e-14 as you suggested. However, the results did not change much compared to rtol(A) = rtol(S) = rtol(\hat{S}).


Additionally, I checked the identity preconditioner instead of \hat{S}. In this case, the method firstly seems to converge when reaching rtol(S)=1e-6, but then, with a further decrease of rtol(S), it begins to diverge slightly though.


I will try generalised EVP.


Thanks,

Vlad


________________________________
От: Barry Smith <bsmith at petsc.dev>
Отправлено: 10 ноября 2021 г. 17:45:55
Кому: Vladislav Pimanov
Копия: petsc-users at mcs.anl.gov
Тема: Re: [petsc-users] How to compute the condition number of SchurComplementMat preconditioned with PCSHELL.


P is a diffusion matrix, which itself is inverted by KSPCG)

This worries me. Unless solved to full precision the action of solving with CG is not a linear operator in the input variable b, this means that the action of your Schur complement is not a linear operator and so iterative eigenvalue algorithms may not work correctly (even if the linear system seems to be converging).

I suggest you try a KSP tolerance of 10^-14 for the inner KSP solve when you attempt the eigenvalue computation.

Barry


On Nov 10, 2021, at 9:09 AM, Vladislav Pimanov <Vladislav.Pimanov at skoltech.ru<mailto:Vladislav.Pimanov at skoltech.ru>> wrote:


Great thanks, Matt!

The second option is what I was looking for.


Best Regards,

Vlad

________________________________
От: Matthew Knepley <knepley at gmail.com<mailto:knepley at gmail.com>>
Отправлено: 10 ноября 2021 г. 16:45:00
Кому: Vladislav Pimanov
Копия: petsc-users at mcs.anl.gov<mailto:petsc-users at mcs.anl.gov>
Тема: Re: [petsc-users] How to compute the condition number of SchurComplementMat preconditioned with PCSHELL.

On Wed, Nov 10, 2021 at 8:42 AM Vladislav Pimanov <Vladislav.Pimanov at skoltech.ru<mailto:Vladislav.Pimanov at skoltech.ru>> wrote:

Dear PETSc community,


I wonder if you could give me a hint on how to compute the condition number of a preconditioned matrix in a proper way.

I have a MatSchurComplement matrix S and a preconditioner P of the type PCSHELL (P is a diffusion matrix, which itself is inverted by KSPCG).

I tried to compute the condition number of P^{-1}S "for free" during the outer PCG procedure using KSPComputeExtremeSingularValues() routine.

Unfortunately, \sigma_min does not converge even if the solution is computed with very high precision.

I also looked at SLEPc interface, but did not realised how PC should be included.

You can do this at least two ways:

  1) Make a MatShell for P^{-1} S. This is easy, but you will not be able to use any factorization-type PC on that matrix.

  2) Solve instead the generalized EVP, S x = \lambda P x. Since you already have P^{-1}, this should work well.

  Thanks,

     Matt

Thanks!


Sincerely,

Vladislav Pimanov



--
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/<http://www.cse.buffalo.edu/~knepley/>

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