[petsc-dev] Multigrid is confusing

[Jed Brown]

On Fri, May 25, 2012 at 9:06 AM, Mark F. Adams <mark.adams at columbia.edu>wrote:

> On May 25, 2012, at 9:42 AM, Jed Brown wrote:
>
> The high end of the GS preconditioned operator is still high frequency. If
> it wasn't, then GS would be a spectrally equivalent preconditioner.
>
>
> Huh?  If I damp Jacobi on the 3-point stencil with 0.5 then the high
> frequency is _not_ the "high end of the preconditioned operator". It is
> asymptotically 0. Does that mean it is spectrally equivalent?
>

When I said "high" frequency, I didn't mean "highest" frequency.

The low end of the spectrum (that you can't capture) is relatively
unperturbed by local smoothers.

So let's look at a damped Jacobi preconditioner. Suppose D =
[diag(A)]^{-1}. If you weight it by w=0.5 or whatever, the Chebyshev(2)
error propagation operator still looks like

(I - a w D A) (I - b w D A)

where a and b come from the target interval and we build eigenvalue
estimates using K = w D A, so we'll produce exactly the same polynomial as
w=1.

We need better visualization for modes, but if the preconditioned operator
K = P^{-1}A has maximum eigenvalue of 1, the second order Chebyshev
polynomial targeting [0.1, 1.1] is about (1 - 0.25 K) (1 - 0.95 K). Thus,
if P^{-1} perfectly corrects the high energy mode, we will use more than
0.95 of that correction.


Please correct the above reasoning if I've messed up.
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