[petsc-dev] How do you get RIchardson?

[Barry Smith]

On Sep 16, 2011, at 4:13 PM, Vijay S. Mahadevan wrote:

> I've always understood nonlinear Richardson to solve a problem A(x) x
> = b as using a linearization and reformulating as
> 
> A(x_n) \delta x_n+1 = r_n,  where r_n = b - A(x_n) x_n, \delta x_n+1 =
> x_n+1 - x_n
> 
> In essence, when A(x_n) contains the exact A evaluated at x_n, it
> simplifies to the standard Newton iteration.

   Yes, this is another valid "generalization" of nonlinear Richardson but it does assume a particular form of the algebraic equations that we don't need.

   But this is actually never Newton: if F(x) = A(x) x - b then J_F(x) = A(x) + J_A(x)x  so Just using A(x_n) will never give you Newton. But yes is commonly done because it doesn't require the horrible computation of J_A(x). In fact I was at a recent Ph D. defense where this was done (and in that case was much slower than doing Newton). 

   Barry


> But when A contains only
> parts of the true A, I understand it to be a nonlinear fixed point
> iteration. This is quite often done with multiphysics problems where
> say with two physics nonlinearly coupled to each other, the true
> jacobian operator (exact newton) is
> 
> A = [ W X ;
>        Y Z ;]
> 
> But with A = [ W 0;
>                     0  Z;]
> 
> it still converges, conditionally to the same solution as exact
> newton. Variations for A yield different rates of convergence. When
> A=1, you get the classical Picard iteration that Matt mentioned (?).
> 
> I like this formulation because it allows the control of including the
> stiff physics and using other algebraic/physics-based preconditioners
> on top of that. I am not sure if this is the standard way of writing
> out nonlinear Richardson or Picard and sorry for adding to the
> confusion ! Just my 2 cents.
> 
> Vijay
> 
> On Fri, Sep 16, 2011 at 8:35 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
>> On Fri, Sep 16, 2011 at 22:14, Matthew Knepley <knepley at gmail.com> wrote:
>>> 
>>> Water Resources is your standard for mathematical terminology?
>> 
>> It's the whole first page of results for each query.
>> More seriously though, what is the problem with
>> x_{n+1} = A(x_n)^{-1} b
>> being a valid fixed-point iteration?