[petsc-dev] Multigrid is confusing

[Jed Brown]

The high end of the GS preconditioned operator is still high frequency. If
it wasn't, then GS would be a spectrally equivalent preconditioner.

There is work on optimizing polynomials for damping on a chosen part of the
spectrum. I think what we really want is to take estimates of a few
extremal eigenvalues and construct a polynomial that damps them.
On May 25, 2012 8:34 AM, "Mark F. Adams" <mark.adams at columbia.edu> wrote:

> Barry,
>
> The trick with Cheby is that it exploits special knowledge: the spectrum
> of the error reduction with an additive (diagonal) and Richardson -- on the
> 5-point stencil of the Laplacian -- is the worst (in the part of the
> spectrum that the smoother has to worry about) at the high end.  2) it is
> cheap to compute the highest eigenvalue and 3) Cheby is made for
> eigenvalues (and you only need the high end so the highest is fine and you
> can makeup the lowest).
>
> The spectrum for multiplicative is quite different (I think. I've never
> actually seen it) so I would expect to see worse performance with Cheby/GS.
>
> This trick works for elasticity but higher order discretizations can cause
> problems.  Likewise, as Jed points out, unsymmetric are another issue.
>  Cheby is a heuristic for the damping for unsymmetric problems and you are
> really on your own.  Note, people have done work on Cheby for unsymmetric
> problems.
>
> Mark
>
> On May 25, 2012, at 8:35 AM, Barry Smith wrote:
>
> >
> > On May 24, 2012, at 10:35 PM, Jed Brown wrote:
> >
> >> On Thu, May 24, 2012 at 10:19 PM, Barry Smith <bsmith at mcs.anl.gov>
> wrote:
> >>
> >> I think I've fixed PCPre/PostSolve_Eisenstat() to work properly with
> kspest but seems to be with ex19 a much worse smoother than SOR (both with
> Cheby). I'm not sure why. With SOR we are essentially doing a few Chebyshev
> iterations with
> >> (U+D)^-1 (L+D)^-1 A  x  = b while with Eisenstat it is (L+D)^-1 A (U +
> D)^-1 y = (L+D)^-1 b.  Well no time to think about it now.
> >>
> >>  Mark,
> >>
> >>         Has anyone looked at using Cheby Eisentat smoothing as opposed
> to Cheby SOR smoothing?
> >>
> >> Aren't they supposed to be the same?
> >
> >  Well one is left preconditioning while the other is "split"
> preconditioning so iterates could be different.   There might also some
> difference with diagonal scalings?
> >
> >   Barry
> >
> >>
> >>
> >>   Barry
> >>
> >> On May 24, 2012, at 1:20 PM, Jed Brown wrote:
> >>
> >>> On Wed, May 23, 2012 at 2:52 PM, Jed Brown <jedbrown at mcs.anl.gov>
> wrote:
> >>> On Wed, May 23, 2012 at 2:26 PM, Barry Smith <bsmith at mcs.anl.gov>
> wrote:
> >>>
> >>> Note that you could use -pc_type eisenstat perhaps in this case
> instead. Might save lots of flops?  I've often wondered about doing Mark's
> favorite chebyshev smoother with Eisenstat, seems like it should be a good
> match.
> >>>
> >>> [0]PETSC ERROR: --------------------- Error Message
> ------------------------------------
> >>> [0]PETSC ERROR: No support for this operation for this object type!
> >>> [0]PETSC ERROR: Cannot have different mat and pmat!
> >>>
> >>> Also, I'm having trouble getting Eisenstat to be more than very
> marginally faster than SOR.
> >>>
> >>>
> >>> I think we should later be getting the eigenvalue estimate by applying
> the preconditioned operator to a few random vectors, then orthogonalizing.
> The basic algorithm is to generate a random matrix X (say 5 or 10 columns),
> compute
> >>>
> >>> Y = (P^{-1} A)^q X
> >>>
> >>> where q is 1 or 2 or 3, then compute
> >>>
> >>> Q R = Y
> >>>
> >>> and compute the largest singular value of the small matrix R. The
> orthogonalization can be done in one reduction and all the MatMults can be
> done together. Whenever we manage to implement a MatMMult and PCMApply or
> whatever (names inspired by VecMDot), this will provide a very low
> communication way to get the eigenvalue estimates.
> >>>
> >>>
> >>> I want to turn off norms in Chebyshev by default (they are very
> wasteful), but how should I make -mg_levels_ksp_monitor turn them back on?
> I'm already tired of typing -mg_levels_ksp_norm_type unpreconditioned.
> >>
> >>
> >
> >
>
>
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