Thank you very much for the answer, but some other doubts remain.
Date: Thu, 16 Feb 2012 13:39:11 -0600
From: Matthew Knepley <knepley@gmail.com>
Subject: Re: [petsc-users] Multigrid as a preconditioner
To: PETSc users list <petsc-users@mcs.anl.gov>
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<CAMYG4Gmt_mn_hPVKOrbC452geco3Zz26SF05pGy6t7RPAq+0Zg@mail.gmail.com>
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On Thu, Feb 16, 2012 at 12:54 PM, <coco@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.
I think my explanation was not so clear. What I would like to do is to use a preconditioned Richardson iteration:
x^(n+1) = x^n + P^(-1) (f-A x^n)
Choosing P=A, I should expect to obtain the exact solution at the first iteration. Then, the whole linear system is solved by the preconditioner method that I chose. Is it what Petsc would do?
I am not sure what you mean by "Is it what Petsc would do?". PETSc does what you tell it to do. If you want it
to solve in one iteration, tell it to use LU, -ksp_type richardson -pc_type lu.
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 tried, but unfortunately nothing changed.
Another strange phenomen is that, even with the standard KSP solver (which provides the right solution), if I use the flag -ksp_monitor, nothing is displayed. Is that an indicator of some troubles?
1) You must call KSPSetFromOptions() to use command line options
2) Run with -ksp_monitor -ksp_view and send the results, or there is no way we can know what is happening.
Matt
Thank you in advance.
Armando
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