On May 25, 2012, at 9:42 AM, Jed Brown wrote:
> The high end of the GS preconditioned operator is still high frequency. If it
> wasn't, then GS would be a spectrally equivalent preconditioner.
>
Huh? If I damp Jacobi on the 3-point stencil with 0.5 then the high frequency
is _not_ the "high end of the preconditioned operator". It is asymptotically 0.
Does that mean it is spectrally equivalent?
> 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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