Question regarding PCMG

[TsuyoshiKoyama(berkeley)]

Thank you very much for the quick response. Considering the first comment,
after again looking at the Richardson KSP code  I observe the following.
Given the precondioned system,

PAx = Pb

the iteration is conducted as,

x_{n+1} = x_{n} + scale*P(b-Ax_{n})

If  one gives, NonZeroInitialGuess what happens is the following,

1) r = (b-A_x{initial guess})
2) Do the following maxit times,

    x_{new} = x_{old} + scale*Pr
    r = b - x_{new}

Thus if one sets the preconditioner Gauss-Seidel as P_{GS}, then one
iteration produces the following value of x

x_{result} = x_{initial guess} + scale*P_{GS}(b-Ax_{initial guess})

when one desires in a Gauss-Seidel smoother,

x_{result} = x_{initial guess} + (L+D)^{-1}(b - Ux_{initial guess})

where, A = L(strict lower) + D+ U(strict upper)

Does this mean that somehow Petsc internally modifies Amat in the PC
object so that Amat = P, and PC_Apply multiplies by (L+D)^{-1}? Am
I missing some detail or misunderstanding the behavior of PC?

As for the second case, the reason I might not select a coarse solver is 
since I
am trying to solve a Helmholtz equation and the coarse grid solve might not 
be
correct enough to apply. Thus, you may just point out that the coarse grid 
should
be large enough to resolve the so called low frequency error and that if you 
do so
you should still be able to solve direct, but I was curious on testing 
different types
of operations at the coarsest level.

In both of the comments I know I can get around them by changing(commenting 
out) the corresponding source code lines(I assume that this effect will be 
minimal though I do not
know how much) but I was wondering if some other people had made similar 
remarks
or questions.

Again, thank you very much.

Sincerely,

Tsuyoshi Koyama

----- Original Message ----- 
From: "Matthew Knepley" <knepley at gmail.com>
To: <petsc-users at mcs.anl.gov>
Sent: Tuesday, August 14, 2007 1:11 AM
Subject: Re: Question regarding PCMG


> On 8/13/07, TsuyoshiKoyama(berkeley) <tkoyama at berkeley.edu> wrote:
>> Dear petsc-users,
>>
>> I am currently trying to use the PCMG object and
>> had some questions with the preconditioner.
>>
>> ---------------------------------------------------------
>> 1. In this multigrid preconditioner, the default KSP smoother seems
>> to be GMRES. If one wanted to use the Gauss-Seidel
>> iteration as a smoother what would one do?
>>
>> The solution that I thought feasible is to use the PCGauss-Seidel
>> and set this into the KSP smoother, and set the Krylov solver
>> type to KSPPreonly. Since we are using a Gauss-Seidel smoothing
>> and in the case that we include post-smoothing, the smoother must
>> be able to incorporate a nonzero initial guess. Thus one would think
>> of setting the flag for KSPSetInitialGuessNonzero. Unfortunately the
>> combination of,
>>
>> -KSPPreonly
>> -KSPSetInitialGuessNonzero
>>
>> is rejected in /src/ksp/ksp/impls/preonly/preonly.c line:24. Here it
>> advises one to use a Richardson but that is not what I would like.
>
> Actually, I believe Richardson is exactly what you want. It is just the 
> simple
> update.
>
>> 2. In the setting of the coarse grid KSP smoother(solver) in PCMG, in the
>> function
>> PCSetUp_MG in file /src/ksp/pc/impls/mg/mg.c line:474 there is
>> a setting that requires one to only be able to use lu, redundant, or
>> cholesky as the solver in the case of preonly. This means that one would
>> not again be able to use the type of solver that I have stated above,
>> a Gauss-Seidel solver or any other user-specified stationary
>> solver with KSPPreonly.
>> ---------------------------------------------------------
>
> I do not really understand what you want here. Can you give me more 
> explanation?
> The coarse problem by default is solved exactly.
>
>  Thanks,
>
>    Matt
>
>> If you have any comments or tricks to get by this, it would be very
>> helpful.
>>
>> Sincerely,
>>
>> -Tsuyoshi Koyama
> -- 
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which
> their experiments lead.
> -- Norbert Wiener
>