On Sun, Mar 3, 2013 at 5:15 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
> 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). > Form the explanation in this paper, as far as I understand it, deflation is just used to augment the Krylov space, so why not just use LGMRES, sticking in your approximations to the eigenspace? Matt > 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. >> > > > -- What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead. -- Norbert Wiener -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20130303/57ca890a/attachment.html>
