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

[Matthew Knepley]

On Mon, Jun 26, 2017 at 8:37 PM, Jason Lefley <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 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 looked at some other code that uses geometric multi-grid to solve the
> same problem and the authors assumed a uniform cell width and subsequently
> factored out the cell width and moved it to the right hand side. I did that
> in our code (using scale = 1 rather than 1/h^2) and we see better
> convergence with the mg pre-conditioner:
>
> $ 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: 512.
> right hand side infinity norm: 0.999097
>   0 KSP preconditioned resid norm 3.434682678336e+05 true resid norm
> 5.120000000000e+02 ||r(i)||/||b|| 1.000000000000e+00
>   1 KSP preconditioned resid norm 7.305460801248e+04 true resid norm
> 1.633308409885e+02 ||r(i)||/||b|| 3.190055488056e-01
>   2 KSP preconditioned resid norm 6.656705346262e+03 true resid norm
> 7.942064132554e+01 ||r(i)||/||b|| 1.551184400890e-01
>   3 KSP preconditioned resid norm 3.523570204496e+03 true resid norm
> 4.046777759638e+01 ||r(i)||/||b|| 7.903862811794e-02
>   4 KSP preconditioned resid norm 9.123967354323e+03 true resid norm
> 2.445432977350e+01 ||r(i)||/||b|| 4.776236283887e-02
>   5 KSP preconditioned resid norm 9.811672802436e+02 true resid norm
> 1.556881108574e+01 ||r(i)||/||b|| 3.040783415183e-02
>   6 KSP preconditioned resid norm 1.106193887270e+03 true resid norm
> 8.752912569969e+00 ||r(i)||/||b|| 1.709553236322e-02
>   7 KSP preconditioned resid norm 3.411263151853e+02 true resid norm
> 4.817172861959e+00 ||r(i)||/||b|| 9.408540746014e-03
>   8 KSP preconditioned resid norm 1.129663122476e+02 true resid norm
> 2.051711481120e+00 ||r(i)||/||b|| 4.007248986563e-03
>   9 KSP preconditioned resid norm 7.776030229135e+01 true resid norm
> 1.092336734730e+00 ||r(i)||/||b|| 2.133470185019e-03
>  10 KSP preconditioned resid norm 3.900236414632e+01 true resid norm
> 4.662658178376e-01 ||r(i)||/||b|| 9.106754254641e-04
>  11 KSP preconditioned resid norm 2.884248061867e+01 true resid norm
> 2.584775749590e-01 ||r(i)||/||b|| 5.048390135919e-04
>  12 KSP preconditioned resid norm 1.275086146987e+01 true resid norm
> 1.183721340034e-01 ||r(i)||/||b|| 2.311955742253e-04
>  13 KSP preconditioned resid norm 3.378721119782e+00 true resid norm
> 5.841425568821e-02 ||r(i)||/||b|| 1.140903431410e-04
> Linear solve converged due to CONVERGED_RTOL iterations 13
> KSP final norm of residual 0.0584143
> Residual 2 norm 0.0584143
> Residual infinity norm 0.000458905
>
> While this looks much better than our previous attempts, I think we might
> end up using the algebraic approach for generating the intermediate
> operators (either gamg or mg with -pc_mg_galerkin) since we want to support
> specification of boundary conditions other than just at the extents of the
> domain and it’s not clear how to handle the coarsening when boundary cells
> and non-boundary cells are combined into a single cell on the coarser grid.
>
> I’d still like to understand the renormalization you mentioned. Do you
> know of any resources that discuss it?
>
> Thanks
>



-- 
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

http://www.caam.rice.edu/~mk51/
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