[petsc-users] BiCGSTAB for general use

[Jed Brown]

On Wed, Aug 17, 2011 at 10:48, Paul Anton Letnes <
paul.anton.letnes at gmail.com> wrote:

> > Let's start with a scatter plot of the eigenvalues. Can you do a problem
> that is representative of the physics in less than, say, 1000 degrees of
> freedom? If so, I would just use Matlab (or Octave, etc).  You want to be
> able to plot the eigenvector associated with a chosen eigenvalue in some way
> that is meaningful to you. We want to see if the wavelength (in terms of the
> variables you are discretizing over) of the modes has some useful
> correlation with the size of the associated eigenvalues. If so, we may be
> able to build some sort of multigrid preconditioner.
>
> 1000 degrees of freedom is a bit little. I took a 4608 x 4608 matrix and
> plotted its eigenvalues as a scatterplot in the complex plane, as well as
> the magnitude of the eigenvalues. See attachments. I can give out better
> quality plots off-list.
>

This looks tricky. It certainly doesn't have the nice structure of a
second-kind integral operator with compact kernel (or even first-kind, which
would be the limit of vanishing regularization). If we're going to solve
this efficiently, we probably need either:

1. A sparse system that approximates this one. This is probably unlikely,
but you might have more insight.

2. A transformation that exposes sparsity. Usually these are hierarchical.
Are there any fast transforms that can be used for your operator (e.g. FMM,
H-matrices)?


> I'm really not sure as to how one can visualize eigenvectors with 4608
> elements...
>

Your independent variables discretize a 2D space of angles, right? Then try
a 2D color plot.


> I have not thought too much about the physical meaning of eigenvectors and
> eigenvalues. In fact, even this system is too small to be of physical
> interest, so I'm not sure what I'd get out of it, to be honest. I suppose
> some eigenvectors might be related to surface plasmon polaritons, and one
> eigenvector is probably related to the specular ("mirror-like") peak.
>

For a problem like this, singular values might be more significant. You
could plot the right and left singular vectors which would (if I understand
the problem correctly correspond to incident and outgoing waveforms. It is
worth trying -ksp_type cgne (conjugate gradients on the normal equations) in
case the singular values are better behaved than the eigenvalues.
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