[petsc-dev] Deflated Krylov solvers for PETSc

[Jed Brown]

This paper acknowledges the MG terminology and includes some numerical
examples.

http://dx.doi.org/10.1007/s10915-009-9272-6

Unfortunately, they only solve heterogenous Poisson, for which all the
deflation algorithms look like crude hacks next to MG (which they don't
show results for).

Note that in this paper, all the methods use the coarse operator E = Z^T A
Z where A is the original operator, not a preconditioned operator. That
makes these deflation methods merely V(0,1) or V(1,0) cycles. In
particular, I don't see anything with a coarse operator E = Z^T (M^{-1/2} A
M^{-1/2}) Z or E = W^T (M^{-1} A) W. If this is indeed true, then I think
it's clear that deflation is something that should be implemented as a PC,
perhaps with Z updated by KSP (if we intend to iteratively compute
approximate low eigenvectors).


On Sun, Mar 3, 2013 at 12:52 AM, Jie Chen <jiechen at mcs.anl.gov> wrote:

> Barry,
>
> Putting P_{A} either to the left or to the right of A means the same
> thing: we avoid touching the subspace spanned by W. This is why the
> orthogonality condition b-Ax_0 \perp W is needed. What the condition says
> is that the initial residual should have no components on W, presumably the
> troublesome part of A. Then in a Krylov method, all later residuals have no
> components on W. In other words, the part of the solution on the subspace W
> is already fully computed even in the 0-th step. Getting such an x_0 is not
> difficult; the difficult part is to define/compute W.
>
> When one adds another preconditioner M to the system, presumably W should
> be the troublesome part of AM instead of A (I am always confused about the
> notation M and M^{-1} but it does not affect my reasoning here). In the
> aggressive way W can consist of eigenvectors of the pencil (A,M)
> corresponding to the smallest eigenvalues in magnitude. On the other hand,
> if one already found a good W for A and he/she is lazy and does not want to
> re-figuring out W, then just use the old W. I guess the beauty of the
> deflation theory is that W can be arbitrary but does not depend on A or M.
>
> I sense that you want a preconditioner appearing in (4) and the formula
> for x_0? I will add something there later.
>
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