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

[Jed Brown]

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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