[petsc-dev] Multigrid is confusing

[Jed Brown]

On Fri, May 25, 2012 at 6:57 PM, Mark F. Adams <mark.adams at columbia.edu>wrote:

> Yes, I forgot what we were arguing about there for a minute :)
>
> My thinking was that Cheby _could_ degrade performance over Richardson
> (for G-S in particular).  That is that I could construct a
> smoother/operator for which Cheby would not pick the correct damping (so if
> the PC/operator was damped perfectly Richardson could win).
>
> I'm not sure if that is true now.
>

So you can of course make it worse for some modes, but only by a modest
factor for real eigenvalues, and Cheby generally hits a broader part of the
spectrum. If the preconditioned operator has eigenvalues with large
imaginary components, Cheby can become unstable.

For that case (and similar in the nonlinear context), I would like to be
able to estimate a few eigenvalues and optimize a polynomial smoother that
damps those regions. For example, attached is an unoptimized example of a
fourth order polynomial.

octave> xx = linspace(0,3,30); yy = linspace(-3,3,60); [x,y] =
meshgrid(xx,yy); z = x+1i*y;
contourf(x,y,1-z/(1+2i)).*(1-z/(1-2i)).*(1-z/(1.5-.9i)).*(1-z/(1.5+.9i))),
[.2 .4 .6 .8 1])

Everything in the dark blue region has a damping factor smaller than 0.4,
everything in the light blue region is damped by 0.6. If the extreme
eigenvalues get closer to the real axis, we can find polynomials that damp
better (or reduce the degree of the polynomial).

There is plenty of related work in the ODE literature, but they also care
about accuracy, which doesn't matter at all to us.
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