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. -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20110917/5458caf0/attachment.html>