[petsc-dev] How do you get RIchardson?

[Barry Smith]

On Sep 16, 2011, at 5:31 PM, Matthew Knepley wrote:

> On Fri, Sep 16, 2011 at 5:28 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
> On Sat, Sep 17, 2011 at 00:20, Matthew Knepley <knepley at gmail.com> wrote:
> My definition is in no way a "strict" subset. Define your nonlinear operator to have a solve, and it has what you want.
> 
> You want to solve
> 
> f(x) = 0
> 
> but you have to write
> 
> x = g(x)
> 
> to apply a fixed point method. So you do, e.g.
> 
> x = A(x)^{-1} b
> 
> Now we have to come up with a nonlinear problem f(x) = 0 such that
> 
> x = x - f(x) = A(x)^{-1} b
> 
> Evidently that is f(x) = x - A(x)^{-1} b. Now I have this extra x floating around just so it can be subtracted. Just because you can transform something to make a certain choice general doesn't make it so.
> 
> It most certainly does make it so. You can definitely, in user code, define an F that produces the iteration that you
> want. You cannot write a Newton that does what I want. How hard is that to understand? It is not hard, in fact, but
> this is arguing for the sake of not being wrong.

  I don't understand what you are talking about; what is b?

   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?   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). 

> 
>    Matt
> 
> -- 
> What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.
> -- Norbert Wiener