<div class="gmail_quote">On Wed, Aug 17, 2011 at 10:48, Paul Anton Letnes <span dir="ltr"><<a href="mailto:paul.anton.letnes@gmail.com">paul.anton.letnes@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div class="im">> 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.<br>
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</div>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.<br>
</blockquote><div><br></div><div>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:</div>
<div><br></div><div>1. A sparse system that approximates this one. This is probably unlikely, but you might have more insight.</div><div><br></div><div>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)?</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
I'm really not sure as to how one can visualize eigenvectors with 4608 elements...<br></blockquote><div><br></div><div>Your independent variables discretize a 2D space of angles, right? Then try a 2D color plot. </div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
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.<br>
</blockquote><div><br></div><div>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.</div>
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