Re: [petsc-users] petsc-users Digest, Vol 38, Issue 41
Thank you for the answer.
Date: Fri, 17 Feb 2012 10:11:51 -0600 From: Matthew Knepley <[email protected]> Subject: Re: [petsc-users] Multigrid as a preconditioner To: PETSc users list <[email protected]> Message-ID: <CAMYG4GkKE6doSQ1FgHCSr1deZRYS0kpvLQF8daqv0au==tOgyw@mail.gmail.com> Content-Type: text/plain; charset="iso-8859-1"
On Fri, Feb 17, 2012 at 6:38 AM, <[email protected]> wrote:
Thank you very much for the answer, but some other doubts remain.
Date: Thu, 16 Feb 2012 13:39:11 -0600
From: Matthew Knepley <[email protected]> Subject: Re: [petsc-users] Multigrid as a preconditioner To: PETSc users list <[email protected]> Message-ID: <CAMYG4Gmt_mn_**hPVKOrbC452geco3Zz26SF05pGy6t7** [email protected]<CAMYG4Gmt_mn_hPVKOrbC452geco3Zz26SF05pGy6t7RPAq%[email protected]>
Content-Type: text/plain; charset="iso-8859-1"
On Thu, Feb 16, 2012 at 12:54 PM, <[email protected]> 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.
Indeed I would like to solve the whole linear system by a multigrid approach and not by a lu factorization. Therefore I would like to use -ksp_type richardson -pc_type mg. In this case, the preconditioned problem P^(-1) (f-A x^n) is solved exactly or it performs just a V-cycle iteration? In both cases, since I am using a one-grid multigrid (just for debugging), it should anyway provide the exact solution at the first iteration, but it is not so.
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. I plotted the residual and it decreases so much slowly. It seems like it is using the non-preconditioned Richardson iteration x^(n+1) = x^n + P^(-1) (f-A x^n) with P=I instead of P=A. Thank you. Armando
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
On Fri, Feb 17, 2012 at 1:09 PM, <[email protected]> wrote:
Thank you for the answer.
Date: Fri, 17 Feb 2012 10:11:51 -0600
From: Matthew Knepley <[email protected]> Subject: Re: [petsc-users] Multigrid as a preconditioner To: PETSc users list <[email protected]> Message-ID: <**CAMYG4GkKE6doSQ1FgHCSr1deZRYS0**kpvLQF8daqv0au==tOgyw@mail.** gmail.com <[email protected]>> Content-Type: text/plain; charset="iso-8859-1"
On Fri, Feb 17, 2012 at 6:38 AM, <[email protected]> wrote:
Thank you very much for the answer, but some other doubts remain.
Date: Thu, 16 Feb 2012 13:39:11 -0600
From: Matthew Knepley <[email protected]> Subject: Re: [petsc-users] Multigrid as a preconditioner To: PETSc users list <[email protected]> Message-ID: <CAMYG4Gmt_mn_****hPVKOrbC452geco3Zz26SF05pGy6t7**** [email protected]<CAMYG4**Gmt_mn_**hPVKOrbC452geco3Zz26SF05pGy6t7 **RPAq%[email protected]<CAMYG4Gmt_mn_hPVKOrbC452geco3Zz26SF05pGy6t7RPAq%[email protected]>
Content-Type: text/plain; charset="iso-8859-1"
On Thu, Feb 16, 2012 at 12:54 PM, <[email protected]> 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.
Indeed I would like to solve the whole linear system by a multigrid approach and not by a lu factorization. Therefore I would like to use -ksp_type richardson -pc_type mg. In this case, the preconditioned problem P^(-1) (f-A x^n) is solved exactly or it performs just a V-cycle iteration? In both cases, since I am using a one-grid multigrid (just for debugging), it should anyway provide the exact solution at the first iteration, but it is not so.
Great, however I have no idea what you are actually using if you do not send what I asked for in the last message. Matt
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. I plotted the residual and it decreases so much slowly. It seems like it is using the non-preconditioned Richardson iteration x^(n+1) = x^n + P^(-1) (f-A x^n) with P=I instead of P=A.
Thank you. Armando
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
-- 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
On Fri, Feb 17, 2012 at 14:09, <[email protected]> wrote:
Indeed I would like to solve the whole linear system by a multigrid approach and not by a lu factorization. Therefore I would like to use -ksp_type richardson -pc_type mg. In this case, the preconditioned problem P^(-1) (f-A x^n) is solved exactly or it performs just a V-cycle iteration? In both cases, since I am using a one-grid multigrid (just for debugging), it should anyway provide the exact solution at the first iteration, but it is not so.
-pc_type mg with one level just applies a normal smoother. I've sometimes thought it should do a coarse-level solve instead, but I haven't messed with it. Barry, why doesn't it do a direct solve? In general -pc_type mg does one multigrid cycle (usually a V or W cycle). If you want to use multiple iterations, you can -pc_type ksp -ksp_pc_type mg which would use the default KSP (GMRES) as an iteration, preconditioned by multigrid. The "outer" problem will see the result of this converged iterative solve.
On Feb 17, 2012, at 2:38 PM, Jed Brown wrote:
On Fri, Feb 17, 2012 at 14:09, <[email protected]> wrote: Indeed I would like to solve the whole linear system by a multigrid approach and not by a lu factorization. Therefore I would like to use -ksp_type richardson -pc_type mg. In this case, the preconditioned problem P^(-1) (f-A x^n) is solved exactly or it performs just a V-cycle iteration? In both cases, since I am using a one-grid multigrid (just for debugging), it should anyway provide the exact solution at the first iteration, but it is not so.
-pc_type mg with one level just applies a normal smoother. I've sometimes thought it should do a coarse-level solve instead, but I haven't messed with it. Barry, why doesn't it do a direct solve?
1) Because MG is an accelerator of the basic smoother, MG is not a deccelerator of a direct solver. That is the action of adding a coarser level is suppose to improve the convergence of the solver. 2) Because if you used a direct solver and the user switched from one to two levels they would be dismayed at the worsening of the convergence. If the user ran a large problem on one level it would run out of memory. 3) I don't think there is really a "correct" abstract or practical answer to which it should be (hence my two snide answers above) I am happy with the current default
In general -pc_type mg does one multigrid cycle (usually a V or W cycle). If you want to use multiple iterations, you can
-pc_type ksp -ksp_pc_type mg
which would use the default KSP (GMRES) as an iteration, preconditioned by multigrid. The "outer" problem will see the result of this converged iterative solve.
You can use -pc_mg_multiplicative_cycles 2 to use 2 V or W cycles as the preconditioner etc. Barry
participants (4)
-
Barry Smith -
coco＠dmi.unict.it -
Jed Brown -
Matthew Knepley