[petsc-users] BiCGSTAB for general use

[Jed Brown]

On Fri, Aug 12, 2011 at 10:09, Paul Anton Letnes <
paul.anton.letnes at gmail.com> wrote:

> The problem is a discretized integral equation. It does not quite fall into
> the boundary element category, but it's not too far off, in a sense. I did
> not do any sophisticated analysis of the singular values, but I do know that
> the condition number (largest over smallest singular value) is not too bad.
>

Thanks for the problem description. Is this a second kind integral operator?
Such systems have "compact + identity" structure, which means that they can
be approximated by low-rank perturbations of the identity. It also means
that Krylov methods converge quickly once they pick up the few eigenvalues
that are not tightly clustered near 1. (The "compact" part implies that the
number of such outliers should be independent of the spatial resolution in
your discretization.)

If you have a fast way to apply the operator (and note that floating point
units are currently 20x to 50x faster than memory bandwidth for
matrix-vector products), even unpreconditioned Krylov methods may be able to
solve your problem well. You can put your algorithm for applying the matrix
inside a MatShell so that all the Krylov methods will work with it.

http://www.mcs.anl.gov/petsc/petsc-as/snapshots/petsc-dev/docs/manualpages/Mat/MatCreateShell.html#MatCreateShell

If you have a hierarchical discretization, or possibly better, a
hierarchical way to apply your operator, then you may be able to use the
hierarchy to put together a multigrid method. Don't worry about this part
until you have gotten your solver working with a Krylov method, and only
then if (a) the number of iterations is sitll large and (b) you want to
direct a fair amount of research effort to this topic. If you meet both
criteria, write back and we can discuss further.
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