[petsc-dev] sor smoothers

[Jed Brown]

"Mark F. Adams" <mfadams at lbl.gov> writes:
> Some hypre papers have shown that cheb/jacobi is faster for some
> problems but for me robustness trumps this for default solver
> parameters in PETSc.

Richardson/SOR is about the best thing out there in terms of reasonable
local work, low/no setup cost, and reliable convergence.  Cheby/Jacobi
just has trivial fine-grained parallelism, but it's not clear that buys
anything on a CPU.

> Jed's analysis suggests that Eisenstat's method saves almost 50% work
> but needs a specialized matrix to get good flop rates.  Something to
> think about doing … 

Mine was too sloppy, Barry got it right.  Eisenstat is for Cheby/SOR,
however, and doesn't do anything for Richardson.

To speed Richardson/SSOR up with a new matrix format, I think we have to
cache the action of the lower triangular part in the forward sweep so
that the back-sweep can use it, and vice-versa.  With full caching and
triangular residual optimization, I think this brings 2 SSOR iterations
of the down-smoother plus a residual to 2.5 work units in the
down-smooth (zero initial guess) and 3 work units in the up-smooth
(nonzero initial guess).  (This is a strong smoother and frequently, one
SSOR would be enough.)
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