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Dear Barry, </p>
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Let S = B A^{-1} B^T be the Schur complement, and \hat{S} = B diag(A)^{-1} B^T denotes the preconditioner.</p>
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I also tried <span style="font-family: Calibri, Helvetica, sans-serif, serif, EmojiFont; font-size: 16px;"><i>KSPComputeExtremeSingularValues
</i>for</span> 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}). </p>
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<p style="">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
<span>though</span>.</p>
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I will try generalised EVP.</p>
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Thanks,</p>
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Vlad</p>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>От:</b> Barry Smith <bsmith@petsc.dev><br>
<b>Отправлено:</b> 10 ноября 2021 г. 17:45:55<br>
<b>Кому:</b> Vladislav Pimanov<br>
<b>Копия:</b> petsc-users@mcs.anl.gov<br>
<b>Тема:</b> Re: [petsc-users] How to compute the condition number of SchurComplementMat preconditioned with PCSHELL.</font>
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<div style="margin-top: 0px; margin-bottom: 0px;" class="">P is a diffusion matrix, which itself is inverted by <i class="">KSPCG</i>)</div>
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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).
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<div class="">I suggest you try a KSP tolerance of 10^-14 for the inner KSP solve when you attempt the eigenvalue computation.</div>
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<div class="">Barry</div>
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<div class="">On Nov 10, 2021, at 9:09 AM, Vladislav Pimanov <<a href="mailto:Vladislav.Pimanov@skoltech.ru" class="">Vladislav.Pimanov@skoltech.ru</a>> wrote:</div>
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<p class="">Great thanks, Matt!</p>
<p class="">The second option is what I was looking for.</p>
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<p class="">Best Regards,</p>
<p class="">Vlad</p>
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<div id="divRplyFwdMsg" dir="ltr" class=""><font face="Calibri, sans-serif" style="font-size:11pt" class=""><b class="">От:</b> Matthew Knepley <<a href="mailto:knepley@gmail.com" class="">knepley@gmail.com</a>><br class="">
<b class="">Отправлено:</b> 10 ноября 2021 г. 16:45:00<br class="">
<b class="">Кому:</b> Vladislav Pimanov<br class="">
<b class="">Копия:</b> <a href="mailto:petsc-users@mcs.anl.gov" class="">petsc-users@mcs.anl.gov</a><br class="">
<b class="">Тема:</b> Re: [petsc-users] How to compute the condition number of SchurComplementMat preconditioned with PCSHELL.</font>
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<div dir="ltr" class="">On Wed, Nov 10, 2021 at 8:42 AM Vladislav Pimanov <<a href="mailto:Vladislav.Pimanov@skoltech.ru" class="">Vladislav.Pimanov@skoltech.ru</a>> wrote:<br class="">
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<p class="">Dear PETSc community,</p>
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<span class="">I wonder if you could give me a hint on how to compute the condition number of a preconditioned matrix in a proper way.</span>
<p class="">I have a <i class="">MatSchurComplement</i> matrix S and a preconditioner P of the type
<i class="">PCSHELL</i> (P is a diffusion matrix, which itself is inverted by <i class="">
KSPCG</i>).</p>
<p class="">I tried to compute the condition number of P^{-1}S "for free" during the outer PCG procedure using <span class=""><i class="">KSPComputeExtremeSingularValues()</i> routine.</span></p>
<p class="">Unfortunately, \sigma_min does not converge even if the solution is computed with very high precision.</p>
<p class=""><span class="">I also looked at SLEPc interface, but did not realised how PC should be included.</span><br class="">
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<div class="">You can do this at least two ways:</div>
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<div class=""> 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.</div>
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<div class=""> 2) Solve instead the generalized EVP, S x = \lambda P x. Since you already have P^{-1}, this should work well.</div>
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<div class=""> Thanks,</div>
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<div class=""> Matt</div>
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<p class="">Thanks!</p>
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<p class="">Sincerely,</p>
<p class="">Vladislav Pimanov</p>
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<div class="">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br class="">
-- Norbert Wiener</div>
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