MG question

[Matthew Knepley]

On Wed, 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)
>
>  II)
>
>  \nabla \cdot (g(x,y) \nabla\phi(x,y))  = f(x,y)
>
>  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?

Well, you might need to prove things, but I would not worry about that first.
It is so easy to code up, just run everything and see what actually works.
Then sit down and try to show it.

  Matt

>  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
>  >
>
>
>



-- 
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