MG question

[Barry Smith]

On Feb 27, 2008, at 2:22 PM, jens.madsen at risoe.dk wrote:

> Ok
>
> Thanks Matthew and Barry
>
> First I solve 2d boundary value problems of size 512^2 - 2048^2.
>
> Typically either kind of problem(solve for phi)
>
> I) poisson type equation:
>
> \nabla^2 \phi(x,y) = f(x,y)

There is no reason to use GMRES here, use

-ksp_type richardson -mg_levels_pc_type sor -mg_levels_ksp_type  
richardson
should require about 5-10 outter iterations to get reasonable  
convergence
on the norm of the residual.

>
>
> II)
>
> \nabla \cdot (g(x,y) \nabla\phi(x,y))  = f(x,y)
>
     If g(x,y) is smooth and not highly varying again you should not  
need GMRES.
If it is a crazy function than the whole kitchen sink will likely give  
better convergence.

    I do not understand your questions. If you don't need GMRES/CG  
then don't use
it and if you think you might need it just try it and see if it helps.

    Barry

> Successively with new f and g functions
>
>
> Do you know where to read about the smoothing properties of GMRES and
> CG? All refs that I find are only describing smoothing with GS-RB etc.
>
> My vague idea on how a fast solver is to use a (preconditioned ILU?)
> krylov (CG for spd ie. problem I, GMRES for II)) method with  
> additional
> MG preconditioning(GS-RB smoother, Krylov solver on coarsest level)?
>
> As my problems are not that big I fear that I will get no MG speedup  
> if
> I use krylov methods as smoothers?
>
> Kind Regards Jens
>
>
> -----Original Message-----
> From: owner-petsc-users at mcs.anl.gov
> [mailto:owner-petsc-users at mcs.anl.gov] On Behalf Of Barry Smith
> Sent: Wednesday, February 27, 2008 8:49 PM
> To: petsc-users at mcs.anl.gov
> Subject: Re: MG question
>
>
>   The reason we default to these "very strong" (gmres + ILU(0))
> smoothers is robustness, we'd rather have
> the solver "just work" for our users and be a little bit slower than
> have it often fail but be optimal
> for special cases.
>
>    Most of the MG community has a mental block about using Krylov
> methods, this is
> why you find few papers that discuss their use with multigrid. Note
> also that using several iterations
> of GMRES (with or without ILU(0)) is still order n work so you still
> get the optimal convergence of
> mutligrid methods (when they work, of course).
>
>    Barry
>
>
> On Feb 27, 2008, at 1:40 PM, Matthew Knepley wrote:
>
>> On Wed, Feb 27, 2008 at 1:31 PM,  <jens.madsen at risoe.dk> wrote:
>>> Hi
>>>
>>> I hope that this question is not outside the scope of this
>>> mailinglist.
>>>
>>> As far as I understand PETSc uses preconditioned GMRES(or another  
>>> KSP
>>> method) as pre- and postsmoother on all multigrid levels? I was just
>>
>> This is the default. However, you can use any combination of KSP/PC
>> on any
>> given level with options. For instance,
>>
>> -mg_level_ksp_type richardson -mg_level_pc_type sor
>>
>> gives "regulation" MG. We default to GMRES because it is more robust.
>>
>>> wondering why and where in the literature I can read about that
>>> method? I
>>> thought that a fast method would be to use MG (with Gauss-Seidel RB/
>>> zebra
>>> smothers) as a preconditioner for GMRES? I have looked at papers
>>> written by
>>> Oosterlee etc.
>>
>> In order to prove something about GMRES/MG, you would need to prove
>> something
>> about the convergence of GMRES on the operators at each level. Good
>> luck. GMRES
>> is the enemy of all convergence proofs. See paper by Greenbaum,
>> Strakos, & Ptak.
>> If SOR works, great and it is much faster. However, GMRES/ILU(0)  
>> tends
>> to be more
>> robust.
>>
>>  Matt
>>
>>> Kind Regards
>> -- 
>> 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
>>
>
>