[petsc-users] fixed point interations

[Matthew Knepley]

On Sun, Nov 6, 2011 at 5:52 PM, Dominik Szczerba <dominik at itis.ethz.ch>wrote:

> >>> I want to start small by porting a very simple code using fixed point
> >>> iterations as follows: A(x)x = b(x) is approximated as A(x0)x = b(x0),
> >>> then solved by KSP for x, then x0 is updated to x, then repeat until
> >>> convergence.
> >
> > Run the usual "Newton" methods with A(x) in place of the true Jacobian.
>
> When I substitute A(x) into eq. 5.2 I get:
>
> A(x) dx = -F(x) (1)
> A(x) dx = -A(x) x + b(x) (2)
> A(x) dx + A(x) x = b(x) (3)
> A(x) (x+dx) = b(x) (4)
>
> My questions:
>
> * Will the procedure somehow optimally group the two A(x) terms into
> one, as in 3-4? This requires knowledge, will this be efficiently
> handled?
>

There is no grouping. You solve for dx and do a vector addition.


> * I am solving for x+dx, while eq. 5.3 solves for dx. Is this, and
> how, correctly handled? Should I somehow disable the update myself?
>

Do not do any update yourself, just give the correct A at each iteration in
your FormJacobian routine.

   Matt


> Thanks a lot,
> Dominik
>
> > can compute A(x) in the residual
> > F(x) = A(x) x - b(x)
> > and cache it in your user context, then pass it back when asked to
> compute
> > the Jacobian.
> > This runs your algorithm (often called Picard) in "defect correction
> mode",
> > but once you write your equations this way, you can try Newton iteration
> > using -snes_mf_operator.
> >
> >>>
> >>> In the documentation chapter 5 I see all sorts of sophisticated Newton
> >>> type methods, requiring computation of the Jacobian. Is the above
> >>> defined simple method still accessible somehow in Petsc or such
> >>> triviality can only be done by hand? Which one from the existing
> >>> nonlinear solvers would be a closest match both in simplicity and
> >>> robustness (even if slow performance)?
> >>
> >> You want -snes_type nrichardson. All you need is to define the residual.
> >
> > Matt, were the 1000 emails we exchanged over this last month not enough
> to
> > prevent you from spreading misinformation under a different name?
>



-- 
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
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