<div class="gmail_quote">On Sun, Aug 14, 2011 at 06:18, 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 id=":dj">I believe of the first kind - there is. Our approach is to discretize the integral equation. The equations we are "really solving" are the Maxwell equations.<br></div></blockquote><div><br></div><div>
Is this the sort of system you're working with?</div><div><br></div><div><a href="http://link.aps.org/doi/10.1103/PhysRevLett.104.223904">http://link.aps.org/doi/10.1103/PhysRevLett.104.223904</a></div><div><br></div>
<div>Note that the system has the form</div><div><br></div><div>J_H(x_\parallel | \omega) = J_H(x_\parallel | \omega)_{inc} + \int (...) G(x | x') J_H(x_\parallel' | \omega) + \int (...) G(x | x') J_E(x_\parallel' | \omega)</div>
<div><br></div><div>which is the form of a second order integral equation. I assume the incident field J_H(...)_{inc} is known in this equation. If you dropped the term on the left hand side in this equation, you would have a Fredholm integral equation of the first kind to "solve", which is problematic at a mathematical level due to ill-posedness.</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div id=":dj"><div class="im">
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</div>I have downloaded and attempted to use a different BiCGSTAB code. It converges, but only after several hundred (about 400) for a very small (not physically interesting) problem. It would appear that if we are to get good performance, some form of preconditioning is necessary.<br>
</div></blockquote><div><br></div><div>Do the eigenvalues decay quickly? Can you plot some eigenvalues? They should decay rapidly to a positive value like (with appropriate scaling) 1.</div><div><br></div></div>