[petsc-users] Singlar values of the GMRES Hessenberg matrix
Dave Lee
davelee2804 at gmail.com
Mon May 27 06:46:01 CDT 2019
Thanks Matt, so just to clarify:
-- VEC_VV() contains the Krylov subspace vectors (left preconditioned if
PC_LEFT) and not the orthonomalized vectors that make up Q_k?
- if so, is it possible to obtain Q_k?
-- HES(row,col) contains the entries of the Hessenberg matrix corresponding
to the Arnoldi iteration for the preconditioned Krylov vectors (if PC_LEFT)?
Cheers, Dave.
On Mon, May 27, 2019 at 8:50 PM Matthew Knepley <knepley at gmail.com> wrote:
> 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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