I think the number of deflation vectors should not be large. So the K_c here is a small matrix, and whether Y is dense or sparse does not make a big difference. In this regard, deflation is not exactly the same as one V-cycle. For multigrid, you coarsen a grid of size 1,000,000 to 500,000. But for deflation, you reduce 1,000,000 to 10 or 50. Make sense?
Jie ----- Original Message ----- From: "Jed Brown" <[email protected]> To: "For users of the development version of PETSc" <petsc-dev at mcs.anl.gov> Sent: Friday, March 1, 2013 3:07:23 PM Subject: Re: [petsc-dev] Deflated Krylov solvers for PETSc I think the most common deflation approach is to use estimates of the most global eigenvectors. Those are dense, so we can construct the coarse operator as K_c = Y^T * (P^{-1} A) * Y without further ado. If subdomains aggregates are used for the deflation vectors Y, then we'd really like to exploit their sparsity. We would seem to want preconditioner application to a special sort of sparse matrix.
