[petsc-users] SLEPc: smallest eigenvalues

[Varun Hiremath]

Hi Jose,

Sorry, it took me a while to test these settings in the new builds. I am
getting good improvement in performance using the preconditioned solvers,
so thanks for the suggestions! But I have some questions related to the
usage.

We are using SLEPc to solve the acoustic modal eigenvalue problem. Attached
is a simple standalone program that computes acoustic modes in a simple
rectangular box. This program illustrates the general setup I am using,
though here the shell matrix and the preconditioner matrix are the same,
while in my actual program the shell matrix computes A*x without explicitly
forming A, and the preconditioner is a 0th order approximation of A.

In the attached program I have tested both
1) the Krylov-Schur with inexact shift-and-invert (implemented under the
option sinvert);
2) the JD solver with preconditioner (implemented under the option usejd)

Both the solvers seem to work decently, compared to no preconditioning.
This is how I run the two solvers (for a mesh size of 1600x400):
$ ./acoustic_box_test.o -nx 1600 -ny 400 -usejd 1 -deflate 1 -eps_target 0
$ ./acoustic_box_test.o -nx 1600 -ny 400 -sinvert 1 -deflate 1 -eps_target 0
Both finish in about ~10 minutes on my system in serial. JD seems to be
slightly faster and more accurate (for the imaginary part of eigenvalue).
The program also runs in parallel using mpiexec. I use complex builds, as
in my main program the matrix can be complex.

Now here are my questions:
1) For this particular problem type, could you please check if these are
the best settings that one could use? I have tried different combinations
of KSP/PC types e.g. GMRES, GAMG, etc, but BCGSL + BJACOBI seems to work
the best in serial and parallel.

2) When I tested these settings in my main program, for some reason the JD
solver was not converging. After further testing, I found the issue was
related to the setting of "-eps_target 0". I have included "
EPSSetTarget(eps,0.0);" in the program and I assumed this is equivalent to
passing "-eps_target 0" from the command line, but that doesn't seem to be
the case. For instance, if I run the attached program without "-eps_target
0" in the command line then it doesn't converge.
$ ./acoustic_box_test.o -nx 1600 -ny 400 -usejd 1 -deflate 1 -eps_target 0
 the above finishes in about 10 minutes
$ ./acoustic_box_test.o -nx 1600 -ny 400 -usejd 1 -deflate 1
 the above doesn't converge even though "EPSSetTarget(eps,0.0);" is
included in the code

This only seems to affect the JD solver, not the Krylov shift-and-invert
(-sinvert 1) option. So is there any difference between passing "-eps_target
0" from the command line vs using "EPSSetTarget(eps,0.0);" in the code? I
cannot pass any command line arguments in my actual program, so need to set
everything internally.

3) Also, another minor related issue. While using the inexact
shift-and-invert option, I was running into the following error:

""
Missing or incorrect user input
Shift-and-invert requires a target 'which' (see EPSSetWhichEigenpairs), for
instance -st_type sinvert -eps_target 0 -eps_target_magnitude
""

I already have the below two lines in the code:
EPSSetWhichEigenpairs(eps,EPS_SMALLEST_MAGNITUDE);
EPSSetTarget(eps,0.0);

so shouldn't these be enough? If I comment out the first line
"EPSSetWhichEigenpairs", then the code works fine.

I have some more questions regarding setting the preconditioner for a
quadratic eigenvalue problem, which I will ask in a follow-up email.

Thanks for your help!

-Varun


On Thu, Jul 1, 2021 at 5:01 AM Varun Hiremath <varunhiremath at gmail.com>
wrote:

> Thank you very much for these suggestions! We are currently using version
> 3.12, so I'll try to update to the latest version and try your suggestions.
> Let me get back to you, thanks!
>
> On Thu, Jul 1, 2021, 4:45 AM Jose E. Roman <jroman at dsic.upv.es> wrote:
>
>> Then I would try Davidson methods https://doi.org/10.1145/2543696
>> You can also try Krylov-Schur with "inexact" shift-and-invert, for
>> instance, with preconditioned BiCGStab or GMRES, see section 3.4.1 of the
>> users manual.
>>
>> In both cases, you have to pass matrix A in the call to EPSSetOperators()
>> and the preconditioner matrix via STSetPreconditionerMat() - note this
>> function was introduced in version 3.15.
>>
>> Jose
>>
>>
>>
>> > El 1 jul 2021, a las 13:36, Varun Hiremath <varunhiremath at gmail.com>
>> escribió:
>> >
>> > Thanks. I actually do have a 1st order approximation of matrix A, that
>> I can explicitly compute and also invert. Can I use that matrix as
>> preconditioner to speed things up? Is there some example that explains how
>> to setup and call SLEPc for this scenario?
>> >
>> > On Thu, Jul 1, 2021, 4:29 AM Jose E. Roman <jroman at dsic.upv.es> wrote:
>> > For smallest real parts one could adapt ex34.c, but it is going to be
>> costly
>> https://slepc.upv.es/documentation/current/src/eps/tutorials/ex36.c.html
>> > Also, if eigenvalues are clustered around the origin, convergence may
>> still be very slow.
>> >
>> > It is a tough problem, unless you are able to compute a good
>> preconditioner of A (no need to compute the exact inverse).
>> >
>> > Jose
>> >
>> >
>> > > El 1 jul 2021, a las 13:23, Varun Hiremath <varunhiremath at gmail.com>
>> escribió:
>> > >
>> > > I'm solving for the smallest eigenvalues in magnitude. Though is it
>> cheaper to solve smallest in real part, as that might also work in my case?
>> Thanks for your help.
>> > >
>> > > On Thu, Jul 1, 2021, 4:08 AM Jose E. Roman <jroman at dsic.upv.es>
>> wrote:
>> > > Smallest eigenvalue in magnitude or real part?
>> > >
>> > >
>> > > > El 1 jul 2021, a las 11:58, Varun Hiremath <varunhiremath at gmail.com>
>> escribió:
>> > > >
>> > > > Sorry, no both A and B are general sparse matrices (non-hermitian).
>> So is there anything else I could try?
>> > > >
>> > > > On Thu, Jul 1, 2021 at 2:43 AM Jose E. Roman <jroman at dsic.upv.es>
>> wrote:
>> > > > Is the problem symmetric (GHEP)? In that case, you can try LOBPCG
>> on the pair (A,B). But this will likely be slow as well, unless you can
>> provide a good preconditioner.
>> > > >
>> > > > Jose
>> > > >
>> > > >
>> > > > > El 1 jul 2021, a las 11:37, Varun Hiremath <
>> varunhiremath at gmail.com> escribió:
>> > > > >
>> > > > > Hi All,
>> > > > >
>> > > > > I am trying to compute the smallest eigenvalues of a generalized
>> system A*x= lambda*B*x. I don't explicitly know the matrix A (so I am using
>> a shell matrix with a custom matmult function) however, the matrix B is
>> explicitly known so I compute inv(B)*A within the shell matrix and solve
>> inv(B)*A*x = lambda*x.
>> > > > >
>> > > > > To compute the smallest eigenvalues it is recommended to solve
>> the inverted system, but since matrix A is not explicitly known I can't
>> invert the system. Moreover, the size of the system can be really big, and
>> with the default Krylov solver, it is extremely slow. So is there a better
>> way for me to compute the smallest eigenvalues of this system?
>> > > > >
>> > > > > Thanks,
>> > > > > Varun
>> > > >
>> > >
>> >
>>
>>
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