[petsc-users] Singlar values of the GMRES Hessenberg matrix

[Dave Lee]

Thanks Matt, great suggestion.

I did indeed find a transpose error this way. The SVD as reconstructed via
U S V^T now matches the input Hessenberg matrix as derived via the
*HES(row,col) macro, and all the singular values are non-zero. However the
solution to example src/ksp/ksp/examples/tutorials/ex1.c as determined via
the expansion over the singular vectors is still not correct. I suspect I'm
doing something wrong with regards to the expansion over the vec array
VEC_VV(), which I assume are the orthonormal vectors of the Q_k matrix in
the Arnoldi iteration....

Thanks again for your advice, I'll keep digging.

Cheers, Dave.

On Thu, May 23, 2019 at 8:20 PM Matthew Knepley <knepley at gmail.com> wrote:

> On Thu, May 23, 2019 at 5:09 AM Dave Lee via petsc-users <
> petsc-users at mcs.anl.gov> wrote:
>
>> Hi PETSc,
>>
>> I'm trying to add a "hook step" to the SNES trust region solver (at the
>> end of the function: KSPGMRESBuildSoln())
>>
>> I'm testing this using the (linear) example:
>> src/ksp/ksp/examples/tutorials/ex1.c
>> as
>> gdb  --args ./test -snes_mf -snes_type newtontr -ksp_rtol 1.0e-12
>> -snes_stol 1.0e-12 -ksp_converged_reason -snes_converged_reason
>> -ksp_monitor -snes_monitor
>> (Ignore the SNES stuff, this is for when I test nonlinear examples).
>>
>> When I call the LAPACK SVD routine via PETSc as
>> PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_(...))
>> I get the following singular values:
>>
>>   0 KSP Residual norm 7.071067811865e-01
>>   1 KSP Residual norm 3.162277660168e-01
>>   2 KSP Residual norm 1.889822365046e-01
>>   3 KSP Residual norm 1.290994448736e-01
>>   4 KSP Residual norm 9.534625892456e-02
>>   5 KSP Residual norm 8.082545620881e-16
>>
>> 1 0.5 -7.85046e-16 1.17757e-15
>> 0.5 1 0.5 1.7271e-15
>> 0 0.5 1 0.5
>> 0 0 0.5 1
>> 0 0 0 0.5
>>
>> singular values: 2.36264 0.409816 1.97794e-15 6.67632e-16
>>
>> Linear solve converged due to CONVERGED_RTOL iterations 5
>>
>> Where the lines above the singular values are the Hessenberg matrix that
>> I'm doing the SVD on.
>>
>
> First, write out all the SVD matrices you get and make sure that they
> reconstruct the input matrix (that
> you do not have something transposed somewhere).
>
>    Matt
>
>
>> When I build the solution in terms of the leading two right singular
>> vectors (and subsequently the first two orthonormal basis vectors in
>> VECS_VV I get an error norm as:
>> Norm of error 3.16228, Iterations 5
>>
>> My suspicion is that I'm creating the Hessenberg incorrectly, as I would
>> have thought that this problem should have more than two non-zero leading
>> singular values.
>>
>> Within my modified version of the GMRES build solution function
>> (attached) I'm creating this (and passing it to LAPACK as):
>>
>>     nRows = gmres->it+1;
>>     nCols = nRows-1;
>>
>>     ierr = PetscBLASIntCast(nRows,&nRows_blas);CHKERRQ(ierr);
>>     ierr = PetscBLASIntCast(nCols,&nCols_blas);CHKERRQ(ierr);
>>     ierr = PetscBLASIntCast(5*nRows,&lwork);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(5*nRows,&work);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nRows*nCols,&R);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nRows*nCols,&H);CHKERRQ(ierr);
>>     for (jj = 0; jj < nRows; jj++) {
>>       for (ii = 0; ii < nCols; ii++) {
>>         R[jj*nCols+ii] = *HES(jj,ii);
>>       }
>>     }
>>     // Duplicate the Hessenberg matrix as the one passed to the SVD
>> solver is destroyed
>>     for (ii=0; ii<nRows*nCols; ii++) H[ii] = R[ii];
>>
>>     ierr = PetscMalloc1(nRows*nRows,&U);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nCols*nCols,&VT);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nRows*nRows,&UT);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nCols*nCols,&V);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nRows,&p);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nCols,&q);CHKERRQ(ierr);
>>     ierr = PetscMalloc1(nCols,&y);CHKERRQ(ierr);
>>
>>     // Perform an SVD on the Hessenberg matrix - Note: this call destroys
>> the input Hessenberg
>>     ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
>>
>> PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("A","A",&nRows_blas,&nCols_blas,R,&nRows_blas,S,UT,&nRows_blas,V,&nCols_blas,work,&lwork,&lierr));
>>     if (lierr) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in SVD
>> Lapack routine %d",(int)lierr);
>>     ierr = PetscFPTrapPop();CHKERRQ(ierr);
>>
>>     // Find the number of non-zero singular values
>>     for(nnz=0; nnz<nCols; nnz++) {
>>       if(fabs(S[nnz]) < 1.0e-8) break;
>>     }
>>     printf("number of nonzero singular values: %d\n",nnz);
>>
>>     trans(nRows,nRows,UT,U);
>>     trans(nCols,nCols,V,VT);
>>
>>     // Compute p = ||r_0|| U^T e_1
>>     beta = gmres->res_beta;
>>     for (ii=0; ii<nCols; ii++) {
>>       p[ii] = beta*UT[ii*nRows];
>>     }
>>     p[nCols] = 0.0;
>>
>>     // Original GMRES solution (\mu = 0)
>>     for (ii=0; ii<nnz; ii++) {
>>       q[ii] = p[ii]/S[ii];
>>     }
>>
>>     // Expand y in terms of the right singular vectors as y = V q
>>     for (jj=0; jj<nnz; jj++) {
>>       y[jj] = 0.0;
>>       for (ii=0; ii<nCols; ii++) {
>>         y[jj] += V[jj*nCols+ii]*q[ii]; // transpose of the transpose
>>       }
>>     }
>>
>>     // Pass the orthnomalized Krylov vector weights back out
>>     for (ii=0; ii<nnz; ii++) {
>>       nrs[ii] = y[ii];
>>     }
>>
>> I just wanted to check that this is the correct way to extract the
>> Hessenberg from the KSP_GMRES structure, and to pass it to LAPACK, and if
>> so, should I really be expecting only two non-zero singular values in
>> return for this problem?
>>
>> Cheers, Dave.
>>
>
>
> --
> 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/
> <http://www.cse.buffalo.edu/~knepley/>
>
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