[petsc-dev] Multigrid is confusing

[Jed Brown]

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.


>
> 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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