[petsc-dev] rename SNES methods ls, tr etc

[Jed Brown]

On Tue, Dec 4, 2012 at 3:48 PM, Anton Popov <popov at uni-mainz.de> wrote:

> Some info on Picard linearization can be found for example in Chapter 2,
> Volume 2 of the  book by Zienkiewicz & Taylor on Finite Elements. Couple of
> equations are given on page 29 (5-th edition), although they're quite
> unclear. Nevertheless it can be helpful.
>
> The idea is that one approximates total nonlinear solution vector (not
> just defect correction) from a linear system with a secant matrix that
> itself depends on the latest solution, and a fixed (at least on a time
> step) right hand side.
>

So you start with the quasi-linear form

F(u) := A(u) u - b = 0

then we can rewrite the iteration

w = A(u)^{-1} b

in defect-correction form

w = u - A(u)^{-1} F(u)

because

A^{-1} F(u) = A^{-1} (A u - b) = u - A^{-1} b


This means that "Newton with J(u) replaced by A(u)" is actually Picard.
When we work in this defect correction form, we can incrementally add terms
to the linear operator without changing the structure of the algorithm.


> When no satisfactory approximation for solution is given, doing couple of
> Picard steps is a good strategy to start with. Then one can switch to
> Newton with/without line search. In general, the advantage of Picard is
> stability at the expense of linear vs. potentially quadratic convergence
> (for Newton, when exact Jacobian and blah-blah-blah is known).
>
> I know, these are just words, it's a bit difficult to generalize it for
> all possible cases.
>
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