[petsc-users] Richardson with multiplicative MG vs. full MG

[Jed Brown]

Shorthand for this is -pc_mg_type kaskade.

On Mon, Aug 13, 2012 at 1:01 PM, John Fettig <john.fettig at gmail.com> wrote:

> Barry,
>
> Thank you for answering my question.  I have another one for you:  it
> seems the special case of zero pre-smooths is somewhat non-trivial.
> The best I can do is set the pre-smoother to Richardson with PCNONE
> and zero as max_its. However, if you aren't careful in setting
> KSPSetInitialGuessNonzero this can have unexpected results since the
> generic KSPSolve will clobber your solution before it even tries a
> convergence criteria (thus ruling out KSPPREONLY).  It also does a
> couple of unnecessary residual calculations. Would it be reasonable to
> put a zero-iteration special case in KSPSolve so that if you don't
> want any iterations it doesn't actually do anything (no setup, no
> preconditioner, no residual, no scaling, etc.)?
>
> Thanks,
> John
>
> On Thu, Aug 9, 2012 at 6:37 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> >
> >   John,
> >
> > On Aug 9, 2012, at 9:50 AM, John Fettig <john.fettig at gmail.com> wrote:
> >
> >> I am a little confused about what Richardson means.  If you use
> >> multiplicative V-cycle multigrid with Richardson KSP (and no
> >> convergence monitor), it sets the applyrichardson operator to
> >> PCApplyRichardson_MG, which appears to just run V-cycles until
> >> convergence.
> >
> >      Yes, this is correct.
> >
> >>  As far as I can tell, it doesn't ever update according
> >> to the documented
> >>
> >> x^{n+1} = x^{n} + scale*B(b - A x^{n})
> >>
> >       In exact arithmetic it is actually "implicitly" doing exactly this
> update.  It is difficult to see why this is true generally (because B is
> rather complicated for multigrid) but if you consider only two levels with
> a direct solver on the coarse grid and SSOR as the pre and post smooth you
> can write out the formulas and map back and forth between the two forms.
>  The reason for the PCApplyRichardson_ forms is because they are a bit more
> efficient than  separating out the action of B and then doing the update as
> above.
> >
> >
> >> If on the other hand you use full MG, it does update according to the
> >> above formula.  This also happens if you set a convergence monitor.
> >>
> >> I can see how multiplicative V-cycle with Richardson is simply using
> >> multigrid as a solver.  What I don't understand is how full MG with
> >> Richardson is using multigrid as a solver, because it is using the
> >> update formula above in between cycles..  Shouldn't there be a
> >> applyrichardson for full multigrid as well?  If not, why?
> >
> >     I think there could be a applyRichardson for full multigrid but it
> would be kind of complicated and would not benefit much because the amount
> of work in a full multigrid step is much higher so the savings would be a
> much lower percentage than with V cycle.
> >
> >    Barry
> >
> >>
> >> Thanks,
> >> John
> >
>
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