[petsc-dev] Deflated Krylov solvers for PETSc

[Jie Chen]

Jed, I guess you can stick to Z as eigenvectors or you can allow Z to be anything a user defines.

Jie



----- Original Message -----
From: "Jed Brown" <jedbrown at mcs.anl.gov>
To: "For users of the development version of PETSc" <petsc-dev at mcs.anl.gov>
Sent: Sunday, March 3, 2013 4:15:05 PM
Subject: Re: [petsc-dev] Deflated Krylov solvers for PETSc


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).