Jean TSHIMANGA Ilunga : April 13, 2006

On some preconditioning techniques for systems with multiple right-hand sides.

Abstract

We are interested in preconditioning techniques for the solution of symmetric positive definite linear systems with multiple right-hand sides. We consider the case where the right-hand sides are available in sequence and not simultaneously. The main idea in these techniques is to use information produced during the solution of one system to precondition the next one. The information involved is either a set of A-conjugate directions or a set of approximate eigenvectors. Three preconditioning techniques are studied : deflation, spectral preconditioning and LBFGS-like preconditioning. Theoretical results concerning these preconditioners are presented. Numerical experiments on two academic problems taken from data assimilation are shown. These experiments, where the matrix varies slowly, allow us to compare those techniques and give preliminary conclusions as well as perpectives for further work.