On May 25, 2012, at 4:20 PM, Jed Brown wrote:
> On Fri, May 25, 2012 at 3:16 PM, Mark F. Adams <mark.adams at columbia.edu>
> 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:
>
>> 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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