[petsc-dev] How do you get RIchardson?

[Jed Brown]

On Sat, Sep 17, 2011 at 00:50, Barry Smith <bsmith at mcs.anl.gov> wrote:

> If one defines a Picard method as any fixed-point iteration then x^{n+1} =
> x^{n} - J(x^{n})^{-1} F(x^{n}) is a Picard iteration for the equation x = x
> - J(x)^{-1} F(x) in other words Newtons' method is a Picard method; is this
> true? Is Picard algorithm a synonym for fixed point iteration?
>

http://en.wikipedia.org/wiki/Picard_iteration (redirects to "Fixed point
iteration")

Also, Tim Kelley's book describes "fixed point iteration" as "also called
nonlinear Richardson iteration, Picard iteration, or the method of
successive substitution".


>   Regardless we can split SNES into two parts: accelerators (nonlinear
> GMRES, Broyden-type, nonlinear CG) and fixed point methods -- Picard
> (steepest descent, Newton, nonlinear SOR) in the exact same way we do linear
> methods.  But one interesting fact is that none of the "accelerators"
> actually accelerate exact Newton, they will all automatically return the
> most recent result and weight the previous steps with a 0, in a sense Newton
> is an "exact solver" in the same way LU is in exact solver in our KSP/PC
> framework and doesn't benefit from an accelerator; but I in the interest of
> uniformity push LU under the PC instead of having some other special class.
> So far I have not split SNES into these two parts (I know Matt doesn't like
> it and maybe we don't need it).
>

I don't want this either. I think the value of the KSP/PC split (which Matt
doesn't like either) is mostly in documentation. If they were all in the
same bag, new users wouldn't have any idea what to put where, but only a
couple KSPs can handle nonlinear preconditioners, so most combinations
wouldn't make any sense.
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