Question regarding PCMG

[TsuyoshiKoyama(berkeley)]

Concerning the first I understand now.... The equation for
Gauss-Seidel is exactly,

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

So what the PC_{GS} does is to apply (L+D)^{-1} to a vector.

This changes the form of my first question which I will restate in a 
following
email.

Sincerely,

-Tsuyoshi Koyma

----- Original Message ----- 
From: "TsuyoshiKoyama(berkeley)" <tkoyama at berkeley.edu>
To: <petsc-users at mcs.anl.gov>
Sent: Tuesday, August 14, 2007 2:38 AM
Subject: Re: Question regarding PCMG


> 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
>>
>