[petsc-dev] Multigrid is confusing

[Jed Brown]

Now perhaps it would be better to use variable damping inside SOR instead
of just on the outside, but I don't know how to select that.
On May 25, 2012 8:42 AM, "Jed Brown" <five9a2 at gmail.com> 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.
>
> 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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