[petsc-users] BiCGSTAB for general use

[Jed Brown]

On Sun, Aug 14, 2011 at 11:33, Paul Anton Letnes <
paul.anton.letnes at gmail.com> wrote:

> Ingve is my supervisor - so it's a good guess! But, no. BiCGSTAB works well
> for his code, by the way. He gets a massive speedup compared to direct
> solvers.
>

Heh, okay. So it works when the problem is of the kind I've seen before and
the theory I'm familiar with says it should work. Comforting perhaps, but
doesn't solve your problem.


>
> I am working on a different equation called the Reduced Rayleigh Equation
> (RRE) for 2 dimensions, which has not been solved directly before (AFAIK).
>  Have a look here:
>
http://www.tandfonline.com/doi/abs/10.1088/0959-7174/6/3/006
> The form of the solution is given in Eq. 5, and the equation itself is
> given in Eq. 8 with some definitions in Eq. 9-11. As far as I can tell there
> is only 2 terms, so it's a first kind equation. But by all means, feel free
> to correct me on that one. The cited paper uses a completely different
> approach (perturbation theory where 0. order is the flat surface) to solving
> the RRE, by the way. Our approach is more "brute force" - discretize the
> integral on a 2D grid in \vec{q}_\parallel space and solve the linear
> equation system.
>

Hmm, I would have to understand the spectral properties of this kernel
better. Is involves products instead of differences (like the more
conventional Green's function you showed earlier) and is clearly
non-compact, so my earlier comment about trying to invert a compact operator
does not apply.

But you also likely won't get any nice decay in the spectrum with which to
use unpreconditioned Krylov methods.


>
> The idea is that the RRE is less memory intensive than the rigorous
> approach in the paper you cited. However, if we can't avoid the LU
> factorization, the RRE approach will be much more CPU intensive, and of
> lesser interest than we would have hoped...
>
> We have submitted a first paper to ArXiv; it will appear on monday, I
> think.
>
> > Note that the system has the form
> >
> > 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)
> >
> > 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.
>
> But since it's of the second kind, it's better posed, right?
>

Right, because the eigenvalues decay rapidly to 1 instead of decaying
rapidly to 0.


>
> > 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.
> >
> > Do the eigenvalues decay quickly? Can you plot some eigenvalues? They
> should decay rapidly to a positive value like (with appropriate scaling) 1.
>
> I'll have a look at the eigenvalues. Is your suggestion to plot the
> magnitude of the complex eigenvectors as a function of their index, after
> sorting them? I guess that must be what you mean by "plotting".
>

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.
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