On Sep 16, 2011, at 5:36 PM, Jed Brown wrote:
> On Sat, Sep 17, 2011 at 00:28, Barry Smith <bsmith at mcs.anl.gov> wrote:
> 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).
>
> I think Vijay intended to distinguish the A in the solve step from the A in
> the residual. Assume the iteration
>
> x_{n+1} = x_n - J(x_n)^{-1} F(x_n)
>
> Assume in this case that F(x_n) can be written as A(x_n) x_n - b where A(x_n)
> distinguishes some "frozen" coefficients. Usually these are all but those
> variables involved in the highest order derivatives.
>
> Now we define J:
>
> J = F_x(x_n), the true Jacobian. This is Newton
>
> J = A(x_n). This is "Picard with solve", the version most commonly used for
> PDEs. This is also the "Picard linearization" that is discussed in many
> textbooks, especially in CFD.
Should we have support for this in SNES as a particular class since it comes
up fairly often (in certain literature): perhaps for short we could call it
SNESPICARD :-)
This was actually one of my largest concern with the previous SNESPICARD is
lots of people would think it implemented "Picard with solve".
Barry
>
> J = \lambda I. This is gradient descent, aka Richardson or "Matt's Picard".
