[petsc-dev] Bordered systems and low-rank corrections

[Mark F. Adams]

The two actually look very similar.  The community that I learned Uzawa from is very familiar with Sherman-Morrison.   Uzawa might in fact be an iterative S-M ... the wikapedia page does not explain how to recover the solution Y and does not accommodate a non-zero RHS for the constraint equations.  Both of which you'd want to do to be general.

On Nov 4, 2011, at 3:28 PM, Jed Brown wrote:

> On Fri, Nov 4, 2011 at 10:08, Mark F. Adams <mark.adams at columbia.edu> wrote:
> Woodbury does not seem natural (ie, efficient) when A is solved iteratively.  These methods rely on multiple solves with A being almost the same cost as one solve, most of the cost going into the matrix setup (factorization).  This is generally not the case with iterative solvers.  How does Woodbury work with inexact solves?  It looks to me like there are rank-of-B + 2 solves here.  Uzawa solvers (iterate on Schur compliment) seem better -- they work fine with inexact solves for A and you can precondition them easily for these special matrices with explic (D - C diag(A)^-1 B)^-1.  They converge very fast, like one digit per iteration even w/o preconditioning in my experience.
> 
> I think both directions are likely useful. I vaguely recall seeing Woodbury used as a preconditioner where the low rank part was computed using an approximate A. We already have support via PCFieldSplit for the Uzawa-type iteration you describe and for the related full-space iteration.

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