[petsc-users] Multigrid as a preconditioner

[Matthew Knepley]

On Thu, Feb 16, 2012 at 12:54 PM, <coco at dmi.unict.it> wrote:

> Dear list,
>
> I would like to parallelize a multigrid code by Petsc. I do not want to
> use the DMMG infrastructure, since it will be replaced in the next PETSc
> release. Therefore I preferred to use the multigrid as a preconditioner. In
> practice, I use the Richardson iteration, choosing the same matrix of the
> linear system as a preconditioner, so that I think the Richardson iteration
> should converge in only one iteration, and effectively it is like solving
> the whole linear system by the multigrid.
>

Your understanding of the Richardson iteration is flawed. You can consult
Yousef Saad's book for the standard definition and anaysis.


> As a first test, I tried to use a one-grid multigrid (then not a truly
> multigrid). I just set the coarse solver (by a standard KSPSolve), and it
> should be enough, because the multigrid starts already from the coarsest
> grid and then it does not need the smoother and the transfer operators.
> Unfortunately, the iteration scheme (which should be the Richardson
> scheme) converges with a reason=4 (KSP_CONVERGED_ITS) to a wrong solution.
> On the other hand, if I solve the whole problem with the standard KSPSolve
> (then withouth setting the multigrid as a preconditioner ...), it converges
> to the right solution with a reason=2.
>

Yes, give Richardson many more iterations, -ksp_max_it.

   Matt


> I thought that the two methods should be the exactly the same method, and
> I do not understand why they provide different convergence results.
>
> Here is the relevant code:
>
> // Set the matrix of the linear system
> Mat Mcc;
> ierr=MatCreate(PETSC_COMM_**WORLD,&Mcc); CHKERRQ(ierr);
> ierr=MatSetType(Mcc, MATMPIAIJ); CHKERRQ(ierr);
> ierr=MatSetSizes(Mcc,PETSC_**DECIDE,PETSC_DECIDE,1000,1000)**;
> CHKERRQ(ierr);
> ierr=setMatrix(Mcc); //It is a routine that set the values of the matrix
> Mcc
>
> // Set the ksp solver with the multigrid as a preconditioner
>  KSP ksp, KspSolver;
>  ierr = KSPCreate(PETSC_COMM_WORLD,&**ksp);CHKERRQ(ierr);
>  ierr = KSPSetType(ksp,KSPRICHARDSON);
>  ierr = KSPGetPC(ksp,&pc);CHKERRQ(**ierr);
>  ierr = PCSetType(pc,PCMG);CHKERRQ(**ierr);
>  ierr = PCMGSetLevels(pc,1,&PETSC_**COMM_WORLD);CHKERRQ(ierr);
>  ierr = PCMGSetType(pc,PC_MG_**MULTIPLICATIVE);CHKERRQ(ierr);
>  ierr = PCMGGetCoarseSolve(pc,&**kspCoarseSolve);CHKERRQ(ierr);
> ierr = KSPSetOperators(**kspCoarseSolve,Mcc,Mcc,**
> DIFFERENT_NONZERO_PATTERN);**CHKERRQ(ierr);
> ierr = KSPSetTolerances(**kspCoarseSolve,1.e-12,PETSC_**
> DEFAULT,PETSC_DEFAULT,PETSC_**DEFAULT);CHKERRQ(ierr);
> ierr = KSPSetOperators(ksp,Mcc,Mcc,**DIFFERENT_NONZERO_PATTERN);**
> CHKERRQ(ierr);
> ierr = KSPSetTolerances(ksp,1.e-12,**PETSC_DEFAULT,PETSC_DEFAULT,**
> PETSC_DEFAULT);CHKERRQ(ierr);
> ierr = KSPSolve(ksp,RHS,U);CHKERRQ(**ierr);
>
> // Solve with the standard KSPSolve
> KSP ksp1;
> ierr = KSPCreate(PETSC_COMM_WORLD,&**ksp1);CHKERRQ(ierr);
> ierr = KSPSetOperators(ksp1,Mcc,Mcc,**DIFFERENT_NONZERO_PATTERN);**
> CHKERRQ(ierr);
> ierr = KSPSetTolerances(ksp1,1.e-12/(**2*nn123),PETSC_DEFAULT,PETSC_**
> DEFAULT,PETSC_DEFAULT);**CHKERRQ(ierr);
> ierr = KSPSolve(ksp1,RHS,U1);CHKERRQ(**ierr);
>
>
> At the end, the Vector U and U1 are different.
> Thank you.
>
> Best regards,
> Armando
>
>


-- 
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
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