[petsc-users] Solving tridiagonal hermitian generalized eigenvalue problem with SLEPC
Toon Weyens
tweyens at fis.uc3m.es
Mon Aug 18 14:54:25 CDT 2014
Hi, thanks for the answers!
I think I expressed myself wrong: I can indeed get it to work with just
using AIJ matrices, as in example 13. This is the way that I am currently
solving my problem. There are only two issues:
1. memory is indeed important so I would certainly like to decrease it by
one third if possible :-) The goal is to make the simulations as fast and
light as possible to be able to perform parameter studies (on the stability
of MHD configurations).
2. I have played around a little bit with the different solvers but it
appears that the standard method and the Arnoldi with explicit restart
method are the best. Some of the others don't converge and some are slower.
The thing is that in the end the matrices that I use are large but they
have a very easy structure: hermitian tri-diagonal. That's why, I think,
slepc usually converges in a few iterations (correct me if I'm wrong).
The problem is that sometimes, when I consider more grid points, the solver
doesn't work any more because apparently it uses the LU decomposition (not
sure for the matrix A or B in A x = lambda B x) and there is a zero pivot
element (see below for error message). In other words: the matrices become
almost singular. This is a characteristic of the numerical method I think.
Is there any way to fix this by setting a more precise threshold or
something?
Also, there is one more thing: is it possible and useful to use the
non-zero structure of A when defining matrix B? Does this affect perfomance?
Thanks in advance!
Toon
Error messages: identical for KRYLOV-SCHUR and ARNOLDI
[0]PETSC ERROR: --------------------- Error Message
------------------------------------
[0]PETSC ERROR: Detected zero pivot in LU factorization:
see http://www.mcs.anl.gov/petsc/documentation/faq.html#ZeroPivot!
[0]PETSC ERROR: Zero pivot row 19 value 6.57877e-16 tolerance 2.22045e-14!
[0]PETSC ERROR:
------------------------------------------------------------------------
[0]PETSC ERROR: Petsc Release Version 3.4.5, Jun, 29, 2014
[0]PETSC ERROR: See docs/changes/index.html for recent updates.
[0]PETSC ERROR: See docs/faq.html for hints about trouble shooting.
[0]PETSC ERROR: See docs/index.html for manual pages.
[0]PETSC ERROR:
------------------------------------------------------------------------
[0]PETSC ERROR: ./PB3D on a debug-complex named toon-XPS-L501X by toon Mon
Aug 18 21:47:41 2014
[0]PETSC ERROR: Libraries linked from
/opt/petsc/petsc-3.4.5/debug-complex/lib
[0]PETSC ERROR: Configure run at Mon Aug 11 10:08:29 2014
[0]PETSC ERROR: Configure options PETSC_ARCH=debug-complex
--with-scalar-type=complex --with-debugging
[0]PETSC ERROR:
------------------------------------------------------------------------
[0]PETSC ERROR: MatPivotCheck_none() line 589 in
/opt/petsc/petsc-3.4.5/include/petsc-private/matimpl.h
[0]PETSC ERROR: MatPivotCheck() line 608 in
/opt/petsc/petsc-3.4.5/include/petsc-private/matimpl.h
[0]PETSC ERROR: MatLUFactorNumeric_SeqAIJ_Inode() line 1837 in
/opt/petsc/petsc-3.4.5/src/mat/impls/aij/seq/inode.c
[0]PETSC ERROR: MatLUFactorNumeric() line 2889 in
/opt/petsc/petsc-3.4.5/src/mat/interface/matrix.c
[0]PETSC ERROR: PCSetUp_LU() line 152 in
/opt/petsc/petsc-3.4.5/src/ksp/pc/impls/factor/lu/lu.c
[0]PETSC ERROR: PCSetUp() line 890 in
/opt/petsc/petsc-3.4.5/src/ksp/pc/interface/precon.c
[0]PETSC ERROR: KSPSetUp() line 278 in
/opt/petsc/petsc-3.4.5/src/ksp/ksp/interface/itfunc.c
[0]PETSC ERROR: PCSetUp_Redundant() line 170 in
/opt/petsc/petsc-3.4.5/src/ksp/pc/impls/redundant/redundant.c
[0]PETSC ERROR: PCSetUp() line 890 in
/opt/petsc/petsc-3.4.5/src/ksp/pc/interface/precon.c
[0]PETSC ERROR: KSPSetUp() line 278 in
/opt/petsc/petsc-3.4.5/src/ksp/ksp/interface/itfunc.c
[0]PETSC ERROR: STSetUp_Shift() line 126 in
/opt/slepc/slepc-3.4.4/src/st/impls/shift/shift.c
[0]PETSC ERROR: STSetUp() line 294 in
/opt/slepc/slepc-3.4.4/src/st/interface/stsolve.c
[0]PETSC ERROR: EPSSetUp() line 215 in
/opt/slepc/slepc-3.4.4/src/eps/interface/setup.c
[0]PETSC ERROR: EPSSolve() line 90 in
/opt/slepc/slepc-3.4.4/src/eps/interface/solve.c
On 18 August 2014 20:52, Barry Smith <bsmith at mcs.anl.gov> wrote:
>
> I would recommend just using AIJ matrices and not worry about SBAIJ at
> this stage. SBAIJ can save you memory but not operations and unless memory
> is really tight is not likely a useful optimization.
>
> What eigensolvers have you tried and which eigenvalues do you want?
>
> Barry
>
> On Aug 18, 2014, at 5:46 AM, Toon Weyens <tweyens at fis.uc3m.es> wrote:
>
> > Dear all,
> >
> > I am using PETSC and SLEPC to simulate a problem in MHD, described in
> http://scitation.aip.org/content/aip/journal/pop/21/4/10.1063/1.4871859.
> >
> > I have a bit of experience with MPI but not too much with PETSC and
> SLEPC. So after reading both user manuals and also the relevant chapters of
> the PETSC developers manual, I still can't get it to work.
> >
> > The problem that I have to solve is a large generalized eigenvalue
> system where the matrices are both Hermitian by blocks and tridiagonal,
> e.g.:
> >
> > ( A11 A12 0 0 0 ) ( B11 B12 0
> 0 0 )
> > ( A12* A22 A23 0 0 ) ( B12* B22 B23
> 0 0 )
> > ( 0 A23* A33 A34 0 ) = lambda ( 0 B23* B33 B34
> 0 )
> > ( 0 0 A34* A44 A45) ( 0 0
> B34* B44 B45)
> > ( 0 0 A45* A55 0 ) ( 0 0
> B45* B55 0 )
> >
> > where Aii = Aii*, with * the Hermitian conjugate. I apologize for the
> ugly representation.
> >
> > The dimensions of both A and B are around 50 to 100 blocks (as there is
> a block per discretized point) and the blocks themselves can vary from 1 to
> more than 100x100 as well (as they correspond to a spectral decomposition).
> >
> > Now, my question is: how to solve this economically?
> >
> > What I have been trying to do is to make use of the fact that the
> matrices are Hermitian and by using matcreatesbaij and through the
> recommended matcreate, matsettype(matsbaij), etc.
> >
> > Could someone help me out? All help would be greatly appreciated!
> >
> > Thank you in advance,
> > Toon
> > UC3M
>
>
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