[petsc-users] Scaling/Preconditioners for Poisson equation

Filippo Leonardi filippo.leonardi at sam.math.ethz.ch
Mon Sep 29 15:47:15 CDT 2014

@ Barry: It may be that I forgot to set the number of levels for the runs.

New experiment with the following options:

-da_refine 5 -pc_type mg -ksp_monitor -log_summary -pc_mg_type full -ksp_view -
pc_mg_log -pc_mg_levels 5 -pc_mg_galerkin -ksp_monitor_true_residual -

on 128^3, and it looks nice: 

  0 KSP Residual norm 5.584601494955e+01 
  0 KSP preconditioned resid norm 5.584601494955e+01 true resid norm 
1.370259979011e+01 ||r(i)||/||b|| 1.000000000000e+00
  1 KSP Residual norm 9.235021247277e+00 
  1 KSP preconditioned resid norm 9.235021247277e+00 true resid norm 
8.185195475443e-01 ||r(i)||/||b|| 5.973461679404e-02
  2 KSP Residual norm 6.344253555076e-02 
  2 KSP preconditioned resid norm 6.344253555076e-02 true resid norm 
1.108015805992e-01 ||r(i)||/||b|| 8.086172134956e-03
  3 KSP Residual norm 1.084530268454e-03 
  3 KSP preconditioned resid norm 1.084530268454e-03 true resid norm 
3.228589340041e-03 ||r(i)||/||b|| 2.356187431214e-04
  4 KSP Residual norm 2.345341850850e-05 
  4 KSP preconditioned resid norm 2.345341850850e-05 true resid norm 
9.362117433445e-05 ||r(i)||/||b|| 6.832365811489e-06
Linear solve converged due to CONVERGED_RTOL iterations 4

I'll try on more processors. Is it correct what I am doing? If so, is there 
some tweak I am missing.

Any suggestion on optimal number of levels VS number of processors?

Btw, thanks a lot, you are always so helpful.

On Monday 29 September 2014 14:59:49 you wrote:
> Filippo Leonardi <filippo.leonardi at sam.math.ethz.ch> writes:
> > Thank you.
> > 
> > Actually I had the feeling that it wasn't my problem with Bjacobi and CG.
> > 
> > So I'll stick to MG. Problem with MG is that there are a lot of parameters
> > to be tuned, so I leave the defaults (expect I select CG as Krylow
> > method). I post just results for 64^3 and 128^3. Tell me if I'm missing
> > some useful detail. (I get similar results with BoomerAMG).
> > 
> > Time for one KSP iteration (-ksp_type cg -log_summary -pc_mg_galerkin
> > -pc_type mg):
> > 32^3 and 1 proc: 1.01e-1
> > 64^3 and 8 proc: 6.56e-01
> > 128^3 and 64 proc: 1.05e+00
> > Number of PCSetup per KSPSolve:
> > 15
> > 39
> > 65
> Presumably you mean PCApply.  Something is wrong here because this
> iteration count is way too high.  Perhaps your boundary conditions are
> nonsymmetric or interpolation is not compatible with the discretization.
> > With BoomerAMG:
> > stable 8 iterations per KSP but time per iteration greater than PETSc MG
> > and still increases:
> > 64^3:  3.17e+00
> > 128^3: 9.99e+00
> > --> For instance with 64^3 (256 iterations):
> In the first pass with geometric multigrid, don't worry about timing and
> get the iterations figured out.  Are you using a cell-centered or
> vertex-centered discretization.  When you say 128^3, is that counting
> the number of elements or the number of vertices?  Note that if you have
> a vertex-centered discretization, you will want a 129^3 grid.  

Cell-centered, counting elements.

> With
> PCMG, make sure you are getting the number of levels of refinement that
> you expect.

> You should see something like the following (this is 193^3).
> $ mpiexec -n 4 ./ex45 -da_refine 5 -pc_type mg -ksp_monitor -pc_mg_type full
> -mg_levels_ksp_type richardson -mg_levels_pc_type sor -ksp_type richardson
> 0 KSP Residual norm 2.653722249919e+03
>   1 KSP Residual norm 1.019366121923e+02
>   2 KSP Residual norm 2.364558296616e-01
>   3 KSP Residual norm 7.438761746501e-04
> Residual norm 1.47939e-06

> You can actually do better than this by using higher order FMG
> interpolation, by going matrix-free, etc.  For example, HPGMG
> (finite-element or finite-volume, see https://hpgmg.org) will solve more
> than a million equations/second per core.  Is your application really
> solving the constant-coefficient Poisson problem on a Cartesian grid, or
> is that just a test?

I actually just need a cell centered Poisson solver on cartesian grids. 
(various boundary conditions). 

Matrix free you mean AMG (like -pc_mg_galerkin)? Does it reach the same 
scalability as GMG?

> > Using Petsc Release Version 3.3.0, Patch 3, Wed Aug 29 11:26:24 CDT 2012
> And a reminder to please upgrade to the current version of PETSc.

Sadly this is not in my power. I had actually had to rollback all the APIs to 
be able to do this test runs.
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