[petsc-users] Matrix-free Multigrid

[Dave May]

Hi Timothee,

Your thinking is correct -- however it is not as bad as you imagine.

If you want to define a matrix free definition of your operator within
geometric MG,
the Mat is required to support several methods in order to be used in
conjunction with smoother.

For Krylov methods, you definitely need to define MatMult.
Assuming you are happy to use a Jacobi preconditioner, you will need to
implement MatGetDiagonal.

I use MF operators within geometric multigrid (on all levels) and find only
these
two operations are required to support using smoothers such as
{richardson,cheby,cg,gmres}+jacobi.

If wish to define your restriction and prolongation operators in a
matrix-free manner
you will need to additionally define the operation MatMultAdd which is
called from
MatInterpolateAdd()  inside PCMG.  If R = P^T, you will also need to define
MatMultTranspose and MatMultTransposeAdd


Cheers
  Dave


On 29 October 2015 at 08:11, Timothée Nicolas <timothee.nicolas at gmail.com>
wrote:

> Hello all,
>
> Eventually, I would like to use matrix-free methods in my multigrid
> solver, because I have a large problem and matrix memory storage is an
> issue. Since all my matrices represent operators, it is quite
> straightforward to make them matrix-free, and according to the manual, I
> can use multigrid on them. However, depending on the method chosen in KSP,
> different matrix operations are used, which the user has to code by
> himself.
>
> That is, in my understanding, coding the subroutine for MATOP_MULT (which
> in principle is sufficient to describe the matrix) is actually not
> sufficient, because the KSP may want internally to get, e.g., the diagonal
> with MatGetDiagonal, in which case I need a routine to define the operation
> MATOP_GET_DIAGONAL (as far as I understand). This means I have to know all
> the matrix related routines used internally by the KSP (which will depend
> on the ksp method as well as the preconditioner choice), and hard code the
> corresponding routines. After all, one could probably do this, but this is
> really tedious work and I have the feeling it is not the approach intended
> by the developers of PETSc.
>
> Would someone have an advice on how to do this efficiently ? Maybe I am
> totally wrong ? But when I tried to do a KSP with multigrid preconditioning
> on a matrix-free matrix, I got errors complaining about MatGetDiagonal and
> this sort of things.
>
> Best
>
> Timothee
>
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