[petsc-users] Matrix-free generalised eigenvalue problem

[Jose E. Roman]

It is possible to pass a different matrix to build the preconditioner. That is, the shell matrix for B (EPSSetOperators) and an explicit matrix (that approximates B) for the preconditioner. For instance, you can try passing M for building the preconditioner. Since M is an explicit matrix, you can try the default preconditioner (block Jacobi with ILU as local solver) or even a full LU decomposition. The effectiveness of the preconditioner will depend on how the update M+R^H P M P R moves the eigenvalues around.

You can do this with STSetSplitPreconditioner() or STSetPreconditionerMat(). In your case any of them will do.

Jose


> El 17 jul 2023, a las 15:50, Quentin Chevalier <quentin.chevalier at polytechnique.edu> escribió:
> 
> Thank you for this suggestion, I tried to implement that but it's
> proven pretty hard to implement MATOP_GET_DIAGONAL without completely
> tanking performance. After all, B is a shell matrix for a reason : it
> looks like M+R^H P M P R with R itself a shell matrix.
> 
> Allow me to point out that I have no shift. My eigenvalue problem is
> purely about the largest ones out there. Section 8.2 and 3.4.3 led me
> to think that there was a way to avoid computing (or writing a shell
> matrix about it) B^-1... But you seem to stress that there's no way
> around it.
> 
> Quentin
> 
> 
> 
> On Mon, 17 Jul 2023 at 11:56, Jose E. Roman <jroman at dsic.upv.es> wrote:
>> 
>> The B-inner product is independent of the ST operator. See Table 3.2. In generalized eigenproblems you always have an inverse.
>> 
>> If your matrix is diagonally dominant, try implementing the MATOP_GET_DIAGONAL operation and using PCJACOBI. Apart from this, you have to build your own preconditioner.
>> 
>> Jose
>> 
>> 
>>> El 17 jul 2023, a las 11:48, Quentin Chevalier <quentin.chevalier at polytechnique.edu> escribió:
>>> 
>>> Hello Jose,
>>> 
>>> I guess I expected B to not be inverted but instead used as a mass for a problem-specific inner product since I specified GHEP as a problem type. p50 of the same user manual seems to imply that that would indeed be the case. I don't see what problem there would be with using a shell B matrix as a weighting matrix, as long as a mat utility is provided of course.
>>> 
>>> I tried the first approach - I set up my KSP as CG since B is hermitian positive-definite (I made a mistake in my first email), but I'm getting a KSPSolve has not converged, reason DIVERGED_ITS error. I'm letting it run for 1000 iterations already so it seems suspiciously slow for a CG solver.
>>> 
>>> I'm grappling with a shell preconditioner now to try and speed it up, but I'm unsure which one allows for shell matrices.
>>> 
>>> Thank you for your time,
>>> 
>>> Quentin
>>> 
>>> 
>>> On Wed, 12 Jul 2023 at 19:24, Jose E. Roman <jroman at dsic.upv.es> wrote:
>>>> 
>>>> By default, it is solving the problem as B^{-1}*A*x=lambda*x (see chapter on Spectral Transformation). That is why A can be a shell matrix without problem. But B needs to be an explicit matrix in order to compute an LU factorization. If B is also a shell matrix then you should set an iterative solver for the associated KSP (see examples in the chapter).
>>>> 
>>>> An alternative is to create a shell matrix M that computes the action of B^{-1}*A, then pass M to the EPS solver as a standard eigenproblem.
>>>> 
>>>> Jose
>>>> 
>>>> 
>>>>> El 12 jul 2023, a las 19:04, Quentin Chevalier <quentin.chevalier at polytechnique.edu> escribió:
>>>>> 
>>>>> Hello PETSc Users,
>>>>> 
>>>>> I have a generalised eigenvalue problem : Ax= lambda Bx
>>>>> I used to have only A as a matrix-free method, I used mumps and an LU preconditioner, everything worked fine.
>>>>> 
>>>>> Now B is matrix-free as well, and my solver is returning an error : "MatSolverType mumps does not support matrix type python", which is ironic given it seem to handle A quite fine.
>>>>> 
>>>>> I have read in the user manual here that there some methods may require additional methods to be supplied for B like MATOP_GET_DIAGONAL but it's unclear to me exactly what I should be implementing and what is the best solver for my case.
>>>>> 
>>>>> A is hermitian, B is hermitian positive but not positive-definite or real. Therefore I have specified a GHEP problem type to the EPS object.
>>>>> 
>>>>> I use PETSc in complex mode through the petsc4py bridge.
>>>>> 
>>>>> Any help on how to get EPS to work for a generalised matrix-free case would be welcome. Performance is not a key issue here - I have a tractable high value case on hand.
>>>>> 
>>>>> Thank you for your time,
>>>>> 
>>>>> Quentin
>>>> 
>>