<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Yes your are right, simply scaling the PC will result in scaling the eigenvalues and hence the Cheby factors.<div><br><div><div>On May 25, 2012, at 11:54 AM, Jed Brown wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div class="gmail_quote">On Fri, May 25, 2012 at 9:06 AM, Mark F. Adams <span dir="ltr"><<a href="mailto:mark.adams@columbia.edu" target="_blank">mark.adams@columbia.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div><div>On May 25, 2012, at 9:42 AM, Jed Brown wrote:</div><br><blockquote type="cite"><p>The high end of the GS preconditioned operator is still high frequency. If it wasn't, then GS would be a spectrally equivalent preconditioner.</p>
</blockquote><div><br></div></div><div>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? </div>
</blockquote></div><br><div>When I said "high" frequency, I didn't mean "highest" frequency.</div><div><br></div><div>The low end of the spectrum (that you can't capture) is relatively unperturbed by local smoothers.</div>
<div><br></div><div>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</div><div><br></div>
<div>(I - a w D A) (I - b w D A)</div><div><br></div><div>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.</div><div><br>
</div><div>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.</div>
<div><br></div><div><br></div><div>Please correct the above reasoning if I've messed up.</div>
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