CSE Seminar
Michele Benzi, Emory University
DATE: Friday, September 5, 2003
TIME: 3:00 P.M.
PLACE: 2240 DCL
1304 W. Springfield Ave., Urbana, IL
TITLE: Some Techniques for Preconditioning Symmetric Indefinite
Linear Systems
Abstract
In spite of much work on preconditioners in the last several years,
comparatively little attention has been devoted to the case of general
large, sparse, symmetric indefinite systems. Problems of this type
arise in several applications, and in particular when eigensolvers
based on shift-and-invert strategies are used to find a few eigenpairs
of large and sparse matrices. When interior eigenvalues are sought,
this leads to shifted matrices of the form $A - \theta I$ (or $A -
\theta B$ for generalized eigenproblems) which are highly indefinite.
Solving linear systems with such matrices by preconditioned Krylov
subspace methods presents a serious challenge. Indefinite symmetric
systems also naturally arise in structural mechanics, stability
analysis (continuation methods for bifurcation points),
electromagnetism (Helmholtz's equation), etc.
The purpose of this talk is to investigate a few approaches to
preconditioning symmetric indefinite matrices. The focus is on the
construction of symmetric indefinite preconditioners to be used in
conjunction with the symmetric QMR algorithm of Freund and Nachtigal.
An attractive feature of this algorithm is that it can take advantage
of symmetry and it can be used with any symmetric preconditioner,
including indefinite ones. Both incomplete factorization
preconditioners and factored sparse approximate inverse techniques will
be considered. In both cases, a pivoting strategy based on the
Bunch-Kaufman algorithm is used to promote numerical stability.
Numerical experiments illustrating the performance of the various
preconditioners will be discussed and compared with results obtained
using a state-of-the art sparse direct solver.
This is joint work with Miroslav Tuma.