On May 25, 2012, at 4:20 PM, Jed Brown wrote:

On Fri, May 25, 2012 at 3:16 PM, Mark F. Adams <[email protected]> wrote: Yes your are right, simply scaling the PC will result in scaling the eigenvalues and hence the Cheby factors.

But that isn't the significant result, it's that even if a preconditioner selectively and perfectly damps the highest eigenvalues (without rescaling other modes), this Cheby configuration will also damp those modes well since the polynomial "keeps" 95% of that damping.

This is a very soft argument Jed, this may not be differential geometry but it is still math ... You bring up a good point, if your PC kills the highest modes then they are gone because you only ever work with the preconditioned system. But you can never kill a mode completely and because Cheby blows up out of its range, on the high end, then even if there a little high stuff left you need to include it in the range of Cheby.

On May 25, 2012, at 11:54 AM, Jed Brown wrote:

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On Fri, May 25, 2012 at 9:06 AM, Mark F. Adams <[email protected]> wrote: On May 25, 2012, at 9:42 AM, Jed Brown wrote:

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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.