On Mar 1, 2013, at 11:39 PM, Jie Chen <jiechen at mcs.anl.gov> wrote:
> Sometimes the convergence rate that is calculated based on the ratio between > lambda_max and lambda_min is rather pessimistic. The distribution of > eigenvalues plays a critical role in convergence. The common argument for the > effectiveness of eigenvector deflation is that "a few extreme eigenvalues > (usually the smallest ones in magnitude) hamper the convergence so deflating > them is helpful." This does not sound clear enough to me, either. But I did > have an experience where after a preconditioner is applied, the spectrum of > the matrix is clustered, except for a few extreme eigenvalues. Deflating them > significantly improves the convergence of CG. So I think the real magic is > not about "50 out of 1 billion", but rather, about how the spectrum of the > matrix changes when you combine deflation with preconditioning. Yes, but you haven't explained yet how preconditioning is combined with deflation. How is that done? Barry > > Jie > > > > > ----- Original Message ----- > From: "Barry Smith" <bsmith at mcs.anl.gov> > To: "For users of the development version of PETSc" <petsc-dev at mcs.anl.gov> > Sent: Friday, March 1, 2013 9:52:25 PM > Subject: Re: [petsc-dev] Deflated Krylov solvers for PETSc > > > This has always been the biggest puzzler for deflation. Say one has a 1 > billion unknown linear system; simple elliptic problem so the eigenvalues are > distributed between lambda_min and lambda_max with the ratio of lambda_max > over lambda_min is pretty big. Now deflate out 50 eigenvalues, so what? how > can deflating out 50 eigenvalues even if they are the most extreme really > affect the convergence rate very much? It is 50 out of 1 billion. Seems too > magical to be believable? > > Barry
