[petsc-users] Issue using multi-grid as a pre-conditioner with KSP for a Poisson problem

Mon Jul 3 10:06:58 CDT 2017

> On Jun 26, 2017, at 7:52 PM, Matthew Knepley <knepley at gmail.com> wrote:
> 
> On Mon, Jun 26, 2017 at 8:37 PM, Jason Lefley <jason.lefley at aclectic.com <mailto:jason.lefley at aclectic.com>> wrote:
>> Okay, when you say a Poisson problem, I assumed you meant
>> 
>>   div grad phi = f
>> 
>> However, now it appears that you have
>> 
>>   div D grad phi = f
>> 
>> Is this true? It would explain your results. Your coarse operator is inaccurate. AMG makes the coarse operator directly
>> from the matrix, so it incorporates coefficient variation. Galerkin projection makes the coarse operator using R A P
>> from your original operator A, and this is accurate enough to get good convergence. So your coefficient representation
>> on the coarse levels is really bad. If you want to use GMG, you need to figure out how to represent the coefficient on
>> coarser levels, which is sometimes called "renormalization".
>> 
>>    Matt
> 
> I believe we are solving the first one. The discretized form we are using is equation 13 in this document: https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/Projects/prjpoisson.pdf <https://www.rsmas.miami.edu/users/miskandarani/Courses/MSC321/Projects/prjpoisson.pdf> Would you clarify why you think we are solving the second equation?
> 
>  Something is wrong. The writeup is just showing the FD Laplacian. Can you take a look at SNES ex5, and let
> me know how your problem differs from that one? There were use GMG and can converge is a few (5-6) iterates,
> and if you use FMG you converge in 1 iterate. In fact, that is in my class notes on the CAAM 519 website. Its possible
> that you have badly scaled boundary values, which can cause convergence to deteriorate.
> 
>   Thanks,
> 
>      Matt
>  

I went through ex5 and some of the other Poisson/multigrid examples again and noticed that they arrange the coefficients in a particular way.

Our original attempt (solver_test.c) and some related codes that solve similar problems use an arrangement like this:

   u(i-1,j,k) - 2u(i,j,k) + u(i+1,j,k)           u(i,j-1,k) - 2u(i,j,k) + u(i,j+1,k)        u(i,j,k-1) - 2u(i,j,k) + u(i,j,k+1) 
----------------------------------------  +   ----------------------------------------  + ----------------------------------------  = f
                dx^2                                           dy^2                                      dz^2

That results in the coefficient matrix containing -2 * (1/dx^2 + 1/dy^2 + 1/dz^2) on the diagonal and 1/dx^2, 1/dy^2 and 1/dz^2 on the off-diagonals. I’ve also looked at some codes that assume h = dx = dy = dz, multiply f by h^2 and then use -6 and 1 for the coefficients in the matrix.

It looks like snes ex5, ksp ex32, and ksp ex34 rearrange the terms like this:

      dy dz (u(i-1,j,k) - 2u(i,j,k) + u(i+1,j,k))           dx dz (u(i,j-1,k) - 2u(i,j,k) + u(i,j+1,k))              dx dy (u(i,j,k-1) - 2u(i,j,k) + u(i,j,k+1))
---------------------------------------------------  +   ---------------------------------------------------  + ---------------------------------------------------  = f dx dy dz
                         dx                                                     dy                                                        dz

I changed our code to use this approach and we observe much better convergence with the mg pre-conditioner. Is this renormalization? Can anyone explain why this change has such an impact on convergence with geometric multigrid as a pre-conditioner? It does not appear that the arrangement of coefficients affects convergence when using conjugate gradient without a pre-conditioner. Here’s output from some runs with the coefficients and right hand side modified as described above:

$ mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale -mg_levels_ksp_max_it 5
right hand side 2 norm: 0.000244141
right hand side infinity norm: 4.76406e-07
  0 KSP preconditioned resid norm 3.578255383614e+00 true resid norm 2.441406250000e-04 ||r(i)||/||b|| 1.000000000000e+00
  1 KSP preconditioned resid norm 1.385321366208e-01 true resid norm 4.207234652404e-05 ||r(i)||/||b|| 1.723283313625e-01
  2 KSP preconditioned resid norm 4.459925861922e-03 true resid norm 1.480495515589e-06 ||r(i)||/||b|| 6.064109631854e-03
  3 KSP preconditioned resid norm 4.311025848794e-04 true resid norm 1.021041953365e-07 ||r(i)||/||b|| 4.182187840984e-04
  4 KSP preconditioned resid norm 1.619865162873e-05 true resid norm 5.438265013849e-09 ||r(i)||/||b|| 2.227513349673e-05
Linear solve converged due to CONVERGED_RTOL iterations 4
KSP final norm of residual 5.43827e-09
Residual 2 norm 5.43827e-09
Residual infinity norm 6.25328e-11

$ mpirun -n 16 ./solver_test -da_grid_x 128 -da_grid_y 128 -da_grid_z 128 -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 5 -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale -mg_levels_ksp_max_it 5 -pc_mg_type full
  0 KSP preconditioned resid norm 3.459879233358e+00 true resid norm 2.441406250000e-04 ||r(i)||/||b|| 1.000000000000e+00
  1 KSP preconditioned resid norm 1.169574216505e-02 true resid norm 4.856676267753e-06 ||r(i)||/||b|| 1.989294599272e-02
  2 KSP preconditioned resid norm 1.158728408668e-04 true resid norm 1.603345697667e-08 ||r(i)||/||b|| 6.567303977645e-05
  3 KSP preconditioned resid norm 6.035498575583e-07 true resid norm 1.613378731540e-10 ||r(i)||/||b|| 6.608399284389e-07
Linear solve converged due to CONVERGED_RTOL iterations 3
KSP final norm of residual 1.61338e-10
Residual 2 norm 1.61338e-10
Residual infinity norm 1.95499e-12

$ mpirun -n 64 ./solver_test -da_grid_x 512 -da_grid_y 512 -da_grid_z 512 -ksp_monitor_true_residual -pc_type mg -ksp_type cg -pc_mg_levels 8 -mg_levels_ksp_type richardson -mg_levels_ksp_richardson_self_scale -mg_levels_ksp_max_it 5 -pc_mg_type full -bc_type neumann
right hand side 2 norm: 3.05176e-05
right hand side infinity norm: 7.45016e-09
  0 KSP preconditioned resid norm 5.330711358065e+01 true resid norm 3.051757812500e-05 ||r(i)||/||b|| 1.000000000000e+00
  1 KSP preconditioned resid norm 4.687628546610e-04 true resid norm 2.452752396888e-08 ||r(i)||/||b|| 8.037179054124e-04
Linear solve converged due to CONVERGED_RTOL iterations 1
KSP final norm of residual 2.45275e-08
Residual 2 norm 2.45275e-08
Residual infinity norm 8.41572e-10
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