[petsc-users] BiCGSTAB for general use

[Paul Anton Letnes]

On 12. aug. 2011, at 21.54, Jed Brown wrote:

> On Fri, Aug 12, 2011 at 12:26, Paul Anton Letnes <paul.anton.letnes at gmail.com> wrote:
> I'm not 100% sure what you mean by "second kind integral operator", but it is a Fredholm equation of the first kind, as far as I understand (my background is physics rather than mathematics).
> 
> Is the thing you're trying to solve with actually of the first kind, not of the second kind?

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.

> http://en.wikipedia.org/wiki/Fredholm_integral_equation
> 
> The distinction is whether there is essentially in whether there is a local part to the equation or not. The issue, as I understand it, is that solving a first-kind integral equation is generally not a stable process because the eigenvalues of the integral operator decay to zero, implying that it is essentially low rank, thus not invertible. Maybe you use some regularization to get a system that is not essentially singular?

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.

So far, we have relied on direct LU factorization (LAPACK/MKL routines) for solving the equation system. In this case, no preconditioning is needed. I have also investigated the singular values of the matrix - the condition number seems to be decent, if not great (if I recall correctly, less than 1000).

Paul.