[petsc-dev] [petsc-users] unreliable AMG in PETSc

[Mark Adams]

>
>
>    Mark,
>
>    How can we insure that that the smoother is SPD? Which are known to be.
> In his case it was Richardson + SSOR on a bunch of processes.
>
>
The problem is stability.  Richardson + SSOR is symmetric and stable for an
SPD matrix (I think) but we do not have a global SSOR, so we really have
Jacobi/SSOR.  The iteration matrix for the smoothers has spectral radius
like || Ident  - omega D^{-1}A|| for damped (omega) Jacobi.  If A is an
M-matrix 0 < || D^{-1}A ||_2 < 2 (barely) and so spectral radius of this
iteration is less that 1.0 and it is stable (slow but stable).  So our
Richardson + SSOR should not blow up on an M-matrix (provably) and works
pretty well in practice, but even a 9-point stencil (2D) is not a M-matrix.

And I think it is hopeless to try to detect indefiniteness, instability in
the "random" (often buggy) matrices that we get tossed to us.  Just too
expensive to do anything reliable.  Symmetry is easy to test for, the
largest eigenvalue is considered easy but can be tricky, as we have seen.
I'm not sure if there are any shortcuts if you just want to know if a
symmetric matrix does not have negative eigenvalues? (I'm not optimistic).
This is all you need but even the relatively easy high eigenvalue can be
hard to make (reasonably) robust and even it has not proof.

Mark
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