[petsc-users] BiCGSTAB for general use

[Paul Anton Letnes]

> 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.

One may exist, but how one can safely obtain one, I have no idea.

> 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)?

Not that I know of. The operator contains integrals over a stochastically rough surface. Due to its random nature, any sparsity structure will depend on each surface realization, so I don't think it's easy to construct a transform that will be helpful in general.

> 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. 

Sort of. That may be a good idea. Perhaps we could use something like that to learn more about the physics anyway.

> 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.

I have so far not plotted or analyzed any singular vectors or eigenvectors. To simulate a physically meaningful system will probably take quite a few CPU hours... I attach a plot of singular values for a small (4608x4608) test matrix. It does not look too great at first glance.

The largest singular value is 20107, the smallest 69.7, giving a ratio of 288 or so.

Paul

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