[petsc-users] SLEPc: smallest eigenvalues

Jose E. Roman jroman at dsic.upv.es
Tue Sep 28 10:09:37 CDT 2021 


> El 28 sept 2021, a las 7:50, Varun Hiremath <varunhiremath at gmail.com> escribió:
> 
> Hi Jose,
> 
> I implemented the LU factorized preconditioner and tested it using PREONLY + LU, but that actually is converging to the wrong eigenvalues, compared to just using BICGS + BJACOBI, or simply computing EPS_SMALLEST_MAGNITUDE without any preconditioning. My preconditioning matrix is only a 1st order approximation, and the off-diagonal terms are not very accurate, so I'm guessing this is why the LU factorization doesn't help much? Nonetheless, using BICGS + BJACOBI with slightly relaxed tolerances seems to be working fine.

If your PCMAT is not an exact inverse, then you have to iterate, i.e. not use KSPPREONLY but KSPBCGS or another.

> 
> I now want to test the same preconditioning idea for a quadratic problem. I am solving a quadratic equation similar to Eqn.(5.1) in the SLEPc manual:
>       (K + lambda*C + lambda^2*M)*x = 0,
> I don't use the PEP package directly, but solve this by linearizing similar to Eqn.(5.3) and calling EPS. Without explicitly forming the full matrix, I just use the block matrix structure as explained in the below example and that works nicely for my case:
> https://slepc.upv.es/documentation/current/src/eps/tutorials/ex9.c.html

Using PEP is generally recommended. The default solver TOAR is memory-efficient and performs less computation than a trivial linearization. In addition, PEP allows you to do scaling, which is often very important to get accurate results in some problems, depending on conditioning.

In your case K is a shell matrix, so things may not be trivial. If I am not wrong, you should be able to use STSetPreconditionerMat() for a PEP, where the preconditioner in this case should be built to approximate Q(sigma), where Q(.) is the quadratic polynomial and sigma is the target.

> 
> In my case, K is not explicitly known, and for linear problems, where C = 0, I am using a 1st order approximation of K as the preconditioner. Now could you please tell me if there is a way to conveniently set the preconditioner for the quadratic problem, which will be of the form [-K 0; 0 I]? Note that K is constructed in parallel (the rows are distributed), so I wasn't sure how to construct this preconditioner matrix which will be compatible with the shell matrix structure that I'm using to define the MatMult function as in ex9.

The shell matrix of ex9.c interleaves the local parts of the first block and the second block. In other words, a process' local part consists of the local rows of the first block followed by the local rows of the second block. In your case, the local rows of K followed by the local rows of the identity (appropriately padded with zeros).

Jose


> 
> Thanks,
> Varun
> 
> On Fri, Sep 24, 2021 at 11:50 PM Varun Hiremath <varunhiremath at gmail.com> wrote:
> Ok, great! I will give that a try, thanks for your help!
> 
> On Fri, Sep 24, 2021 at 11:12 PM Jose E. Roman <jroman at dsic.upv.es> wrote:
> Yes, you can use PCMAT https://petsc.org/release/docs/manualpages/PC/PCMAT.html then pass a preconditioner matrix that performs the inverse via a shell matrix.
> 
> > El 25 sept 2021, a las 8:07, Varun Hiremath <varunhiremath at gmail.com> escribió:
> > 
> > Hi Jose,
> > 
> > Thanks for checking my code and providing suggestions. 
> > 
> > In my particular case, I don't know the matrix A explicitly, I compute A*x in a matrix-free way within a shell matrix, so I can't use any of the direct factorization methods. But just a question regarding your suggestion to compute a (parallel) LU factorization. In our work, we do use MUMPS to compute the parallel factorization. For solving the generalized problem, A*x = lambda*B*x, we are computing inv(B)*A*x within a shell matrix, where factorization of B is computed using MUMPS. (We don't call MUMPS through SLEPc as we have our own MPI wrapper and other user settings to handle.)
> > 
> > So for the preconditioning, instead of using the iterative solvers, can I provide a shell matrix that computes inv(P)*x corrections (where P is the preconditioner matrix) using MUMPS direct solver?
> > 
> > And yes, thanks, #define PETSC_USE_COMPLEX 1 is not needed, it works without it.
> > 
> > Regards,
> > Varun
> > 
> > On Fri, Sep 24, 2021 at 9:14 AM Jose E. Roman <jroman at dsic.upv.es> wrote:
> > If you do
> > $ ./acoustic_matrix_test.o -shell 0 -st_type sinvert -deflate 1
> > then it is using an LU factorization (the default), which is fast.
> > 
> > Use -eps_view to see which solver settings are you using.
> > 
> > BiCGStab with block Jacobi does not work for you matrix, it exceeds the maximum 10000 iterations. So this is not viable unless you can find a better preconditioner for your problem. If not, just using EPS_SMALLEST_MAGNITUDE will be faster.
> > 
> > Computing smallest magnitude eigenvalues is a difficult task. The most robust way is to compute a (parallel) LU factorization if you can afford it.
> > 
> > 
> > A side note: don't add this to your source code
> > #define PETSC_USE_COMPLEX 1
> > This define is taken from PETSc's include files, you should not mess with it. Instead, you probably want to add something like this AFTER #include <slepceps.h>:
> > #if !defined(PETSC_USE_COMPLEX)
> > #error "Requires complex scalars"
> > #endif
> > 
> > Jose
> > 
> > 
> > > El 22 sept 2021, a las 19:38, Varun Hiremath <varunhiremath at gmail.com> escribió:
> > > 
> > > Hi Jose,
> > > 
> > > Thank you, that explains it and my example code works now without specifying "-eps_target 0" in the command line.
> > > 
> > > However, both the Krylov inexact shift-invert and JD solvers are struggling to converge for some of my actual problems. The issue seems to be related to non-symmetric general matrices. I have extracted one such matrix attached here as MatA.gz (size 100k), and have also included a short program that loads this matrix and then computes the smallest eigenvalues as I described earlier.
> > > 
> > > For this matrix, if I compute the eigenvalues directly (without using the shell matrix) using shift-and-invert (as below) then it converges in less than a minute.
> > > $ ./acoustic_matrix_test.o -shell 0 -st_type sinvert -deflate 1
> > > 
> > > However, if I use the shell matrix and use any of the preconditioned solvers JD or Krylov shift-invert (as shown below) with the same matrix as the preconditioner, then they struggle to converge.
> > > $ ./acoustic_matrix_test.o -usejd 1 -deflate 1
> > > $ ./acoustic_matrix_test.o -sinvert 1 -deflate 1
> > > 
> > > Could you please check the attached code and suggest any changes in settings that might help with convergence for these kinds of matrices? I appreciate your help!
> > > 
> > > Thanks,
> > > Varun
> > > 
> > > On Tue, Sep 21, 2021 at 11:14 AM Jose E. Roman <jroman at dsic.upv.es> wrote:
> > > I will have a look at your code when I have more time. Meanwhile, I am answering 3) below...
> > > 
> > > > El 21 sept 2021, a las 0:23, Varun Hiremath <varunhiremath at gmail.com> escribió:
> > > > 
> > > > 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.
> > > 
> > > You should either do
> > > 
> > > EPSSetWhichEigenpairs(eps,EPS_SMALLEST_MAGNITUDE);
> > > 
> > > without shift-and-invert or
> > > 
> > > EPSSetWhichEigenpairs(eps,EPS_TARGET_MAGNITUDE);
> > > EPSSetTarget(eps,0.0);
> > > 
> > > with shift-and-invert. The latter can also be used without shift-and-invert (e.g. in JD).
> > > 
> > > I have to check, but a possible explanation why in your comment above (2) the command-line option -eps_target 0 works differently is that it also sets -eps_target_magnitude if omitted, so to be equivalent in source code you have to call both
> > > EPSSetWhichEigenpairs(eps,EPS_TARGET_MAGNITUDE);
> > > EPSSetTarget(eps,0.0);
> > > 
> > > Jose
> > > 
> > > > 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
> > > > > > > 
> > > > > > 
> > > > > 
> > > > 
> > > > <acoustic_box_test.cpp>
> > > 
> > > <acoustic_matrix_test.cpp><MatA.gz>
> > 
>

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