> 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 -------------- next part -------------- A non-text attachment was scrubbed... Name: svdvals.pdf Type: application/pdf Size: 70433 bytes Desc: not available URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20110822/95821926/attachment-0001.pdf>
