[petsc-users] [SLEPc] Krylov-Schur convergence

[Jose E. Roman]

As Matt points out, the probable cause is that your matrix is not exactly Hermitian. In our implementation of Lanczos we assume symmetry, so if ||A-A^*|| is non-negligible then some information is being discarded, which may confuse Lanczos giving some eigenpair as converged when it is not.

Why doesn't this happen with the basic Lanczos solver? I don't know, probably due to the restart scheme. In symmetric Krylov-Schur, restart is done with eigenvectors of the projected problem (thick-restart Lanczos), while non-symmetric Krylov-Schur uses Schur vectors. In the basic Lanczos, restart is done building a new initial vector explicitly, which is harmless in this case.

Suggestion: either use non-Hermitian Krylov-Schur or symmetrize your matrix as (A+A^*)/2 (or better when contructing it, note that transposing a parallel matrix with 1024 processes requires a lot of communication).

Jose


> El 13 nov 2018, a las 13:56, Matthew Knepley <knepley at gmail.com> escribió:
> 
> On Tue, Nov 13, 2018 at 7:48 AM Ale Foggia via petsc-users <petsc-users at mcs.anl.gov> wrote:
> Thanks Jose for the answer. I tried it an it worked! I send the output of -eps_view again, just in case:
> 
> EPS Object: 1024 MPI processes
>   type: krylovschur
>     50% of basis vectors kept after restart
>     using the locking variant
>   problem type: non-symmetric eigenvalue problem
>   selected portion of the spectrum: smallest real parts
>   number of eigenvalues (nev): 1
>   number of column vectors (ncv): 16
>   maximum dimension of projected problem (mpd): 16
>   maximum number of iterations: 291700777
>   tolerance: 1e-09
>   convergence test: relative to the eigenvalue
> BV Object: 1024 MPI processes
>   type: svec
>   17 columns of global length 2333606220
>   vector orthogonalization method: classical Gram-Schmidt
>   orthogonalization refinement: if needed (eta: 0.7071)
>   block orthogonalization method: GS
>   doing matmult as a single matrix-matrix product
> DS Object: 1024 MPI processes
>   type: nhep
>   parallel operation mode: REDUNDANT
> ST Object: 1024 MPI processes
>   type: shift
>   shift: 0.
>   number of matrices: 1
> 
>               k               ||Ax-kx||/||kx||
>    -----------------       ------------------
>      -15.048025        1.85112e-10
>      -15.047159        3.13104e-10
> 
> Iterations performed 18
> 
> Why does treating it as non-Hermitian help the convergence? Why doesn't this happen with Lanczos? I'm lost :/
> 
> Are you sure your matrix is Hermitian? Lanczos will work for non-Hermitian things (I believe).
> 
>   Thanks,
> 
>      Matt
>  
> 
> Ale
> 
> El mar., 13 nov. 2018 a las 12:34, Jose E. Roman (<jroman at dsic.upv.es>) escribió:
> This is really strange. We cannot say what is going on, everything seems fine.
> Could you try solving the problem as non-Hermitian to see what happens? Just run with -eps_non_hermitian. Depending on the result, we can suggest other things to try.
> Jose
> 
> 
> > El 13 nov 2018, a las 10:58, Ale Foggia via petsc-users <petsc-users at mcs.anl.gov> escribió:
> > 
> > Hello,
> > 
> > I'm using SLEPc to get the smallest real eigenvalue (EPS_SMALLEST_REAL) of a Hermitian problem (EPS_HEP). The linear size of the matrices I'm solving is around 10**9 elements and they are sparse. I've asked a few questions before regarding the same problem setting and you suggested me to use Krylov-Schur (because I was using Lanczos). I tried KS and up to a certain matrix size the convergence (relative to the eigenvalue) is good, it's around 10**-9, like with Lanczos, but when I increase the size I start getting the eigenvalue with only 3 correct digits. I've used the options: -eps_tol 1e-9 -eps_mpd 100 (16 was the default), but the only thing I got is one more eigenvalue with the same big error, and the iterations performed were only 2. Why didn't it do more in order to reach the convergence? Should I set other parameters? I don't know how to work out this problem, can you help me with this please? I send the -eps_view output and the eigenvalues with its errors:
> > 
> > EPS Object: 2048 MPI processes
> >   type: krylovschur
> >     50% of basis vectors kept after restart
> >     using the locking variant
> >   problem type: symmetric eigenvalue problem
> >   selected portion of the spectrum: smallest real parts
> >   number of eigenvalues (nev): 1
> >   number of column vectors (ncv): 101
> >   maximum dimension of projected problem (mpd): 100
> >   maximum number of iterations: 46210024
> >   tolerance: 1e-09
> >   convergence test: relative to the eigenvalue
> > BV Object: 2048 MPI processes
> >   type: svec
> >   102 columns of global length 2333606220
> >   vector orthogonalization method: classical Gram-Schmidt
> >   orthogonalization refinement: if needed (eta: 0.7071)
> >   block orthogonalization method: GS
> >   doing matmult as a single matrix-matrix product
> > DS Object: 2048 MPI processes
> >   type: hep
> >   parallel operation mode: REDUNDANT
> >   solving the problem with: Implicit QR method (_steqr)
> > ST Object: 2048 MPI processes
> >   type: shift
> >   shift: 0.
> >   number of matrices: 1
> > 
> >               k                ||Ax-kx||/||kx||
> >    -----------------        ------------------
> >      -15.093051         0.00323917 (with KS)
> >      -15.087320         0.00265215 (with KS)
> >      -15.048025         8.67204e-09 (with Lanczos)
> > Iterations performed 2
> > 
> > Ale
> 
> 
> 
> -- 
> 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/