[petsc-users] Diagnosing Poisson Solver Behavior

[Matthew Knepley]

On Sat, Oct 10, 2015 at 7:51 PM, K. N. Ramachandran <knram06 at gmail.com>
wrote:

> Hello all,
>
> I am a graduate student pursuing my Master's and I am trying to benchmark
> a previous work by using PETSc for solving Poisson's Equation.
>
> I am starting off with a serial code and I am trying to keep my code
> modular, i.e. I generate the sparse matrix format and send it to PETSc or
> any other solver. So I haven't built my code from the ground up using
> PETSc's native data structures.
>
> I am having trouble understanding the behavior of the solver and would
> like your thoughts or inputs on what I can do better. I have both Dirichlet
> and Neumann boundary conditions and my matrix size (number of rows) is
> around a million but very sparse (~7 nonzeros per row), as can be expected
> from a finite difference discretization of Poisson's equation.
>
> I tried the methods outlined here
> <http://scicomp.stackexchange.com/questions/513/why-is-my-iterative-linear-solver-not-converging?rq=1>
> and here
> <http://scicomp.stackexchange.com/questions/34/how-can-i-estimate-the-condition-number-of-a-large-sparse-matrix-using-petsc>.
> Reverting to a 41^3 grid, I got the approximate condition number (using  -ksp_monitor_singular_value
> -ksp_type gmres -ksp_gmres_restart 1000 -pc_type none) as ~9072, which
> seems pretty large. Higher matrix sizes give a larger condition number.
>
> 1) My best performing solver + preconditioner is bcgs+ilu(0) (on 1e6
> grid) which solves in around 32 seconds, 196 iterations. How do I get a
> fix for what the lower bound on the running time could be?
>

Unfortunately, that is really slow. You can look at this Tutorial (
http://www.mcs.anl.gov/petsc/documentation/tutorials/ParisTutorial.pdf)
slides 125-126 where
I solve elliptic systems using MG for systems with > 1M unknowns in that
amount of time.

You need to use Multigrid. Its the only solver that should be used for this
problem. Anything else is a waste of time.


> 2) Initially -pc_type hypre just Diverged and I was never able to use it.
> Looking at this thread
> <http://lists.mcs.anl.gov/pipermail/petsc-users/2013-October/019127.html>,
> I had tried the options and it no longer diverges, but the residuals reduce
> and hover around a constant value. How do I work with hypre to get a
> useful preconditioner?
>

Something is wrong with your setup. Hypre is a fantastic solver for these
problems. Send the convergence output (-ksp_monitor_true_residual
-ksp_converged_reason -ksp_view).


> Initially I solve Laplace's equation, so the mesh grid size has no effect
> and even when I solve Poisson's equation, the spacing is carried over to
> the RHS, so I am pretty sure the spacing is not affecting the condition
> number calculation.
>

No, the Laplacian has a condition number that grows like h^{-2}. See
earlier slides in that tutorial.

  Thanks,

     Matt


> Hope this helps. Please let me know if you might need more information.
>
> Thanking You,
> K.N.Ramachandran
> Ph: 814-441-4279
>



-- 
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
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