[petsc-users] Singlar values of the GMRES Hessenberg matrix
Matthew Knepley
knepley at gmail.com
Mon May 27 05:50:25 CDT 2019
On Mon, May 27, 2019 at 3:55 AM Dave Lee <davelee2804 at gmail.com> wrote:
> Hi Matt and PETSc.
>
> Thanks again for the advice.
>
> So I think I know what my problem might be. Looking at the comments above
> the function
> KSPInitialResidual()
> in
> src/ksp/ksp/interface/itres.c
> I see that the initial residual, as passed into VEC_VV(0) is the residual
> of the *preconditioned* system (and that the original residual goes
> temporarily to gmres->vecs[1]).
>
> So I'm wondering, is the Hessenberg, as derived via the *HES(row,col) macro
> the Hessenberg for the original Krylov subspace, or the preconditioned
> subspace?
>
Left-preconditioning changes the operator, so you get he Arnoldi subspace
for the transforned operator, starting with a transformed rhs.
Thanks,
Matt
> Secondly, do the vecs within the KSP_GMRES structure, as accessed via
> VEC_VV() correspond to the (preconditioned) Krylov subspace or the
> orthonormalized vectors that make up the matrix Q_k in the Arnoldi
> iteration? This isn't clear to me, and I need to access the vectors in Q_k in
> order to expand the corrected hookstep solution.
>
> Thanks again, Dave.
>
> On Sat, May 25, 2019 at 6:18 PM Dave Lee <davelee2804 at gmail.com> wrote:
>
>> Thanks Matt, this is where I'm adding in my hookstep code.
>>
>> Cheers, Dave.
>>
>> On Fri, May 24, 2019 at 10:49 PM Matthew Knepley <knepley at gmail.com>
>> wrote:
>>
>>> On Fri, May 24, 2019 at 8:38 AM Dave Lee <davelee2804 at gmail.com> wrote:
>>>
>>>> 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....
>>>>
>>>
>>> Here we are building the solution:
>>>
>>>
>>> https://bitbucket.org/petsc/petsc/src/7c23e6aa64ffbff85a2457e1aa154ec3d7f238e3/src/ksp/ksp/impls/gmres/gmres.c#lines-331
>>>
>>> There are some subtleties if you have a nonzero initial guess or a
>>> preconditioner.
>>>
>>> Thanks,
>>>
>>> Matt
>>>
>>>
>>>> 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/>
>>>>>
>>>>
>>>
>>> --
>>> 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/>
>>>
>>
--
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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