[petsc-dev] How do you get RIchardson?

[Barry Smith]

   This will take more research, the references you cite below are too cluttered with other considerations to be definitive.

    In the current "Picard/Richardson" we have two cases 

          x^{n+1}   = x^{n}  - lambda F(x^{n})     the default and then

          x^{n+1}   = x^{n}  +  lambda d^{n}    where d^{n} is obtained by applying any iterative procedure to SNESSolve(x^{n}) resulting in \hat{x}^{n+1} and then computing d^{n} = \hat{x}^{n+1} - x^{n}

    There is an analogy with unpreconditioned Richardson in the linear case where F(x) = Ax - b  hence F(x^{n}) = Ax^{n} - b  and with preconditioned Richardson B(b - Ax^{n}) is equivalent to d^{n} = B(b - Ax^{n}) so one can easily win an argument that what we have implemented is a general "preconditioned" nonlinear Richardson that is a direct generalization of preconditioned Richardson for linear systems.

    What concerns me about calling it Picard is that different people use Picard to mean different things, for example  Jed calls the iterative process for the nonlinear operator div K(u) grad u  Picard when he does solves of  div K(u^{n} grad u^{n+1}, then there is the whole business of Picard for ODEs.  Do they all fit into the case above?


   I admit I've never heard the term Richardson applied for nonlinear equations but I like the fact that we will have similar names for the linear and nonlinear cases: we will have nonlinear Richardson, nonlinear GMRES, nonlinear CG for SNES (all allowing nonlinear preconditioning) to go with the linear Richardson, GMRES, and CG (all allowing preconditioning). Neat and clean (of course we need to clarify everything well in the manual pages, maybe change SNESRICHARSON to SNESNRICHARDSON). 

    I'm willing to put back Picard but only if it makes sense.

    Barry



On Sep 16, 2011, at 8:33 AM, Matthew Knepley wrote:

> http://petsc.cs.iit.edu/petsc/petsc-dev/rev/387e672db72a
> 
> These are cite Picard for this method:
> 
>  http://math.fullerton.edu/mathews/n2003/picarditerationmod.html
>  http://cdsweb.cern.ch/record/419580/files/ext-2000-010.pdf
>  http://books.google.com/books/about/Computational_solution_of_nonlinear_oper.html?id=dexQAAAAMAAJ
>  http://books.google.com/books?id=BVkbypnD0yUC&pg=PA87&lpg=PA87&dq=Terence+tao+picard&source=bl&ots=VQOpQbFsyn&sig=b-KdPdTXtenRi-dBPsLfdhHcPpY&hl=en&ei=1U9zTv2uFungiAKouuWzAg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CD4Q6AEwBg#v=onepage&q&f=false
> 
> Richardson is specifically for linear equations I thought.
> 
>     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