A block preconditioning cost analysis for solving saddle-point linear systems

Abstract

We investigate the cost of preconditioning when solving large sparse saddlepoint linear systems with Krylov subspace methods. To use the block structure of the original matrix, we apply one of two block preconditioners. Algebraic eigenvalue analysis is given for a particular case of the preconditioners. We also give eigenvalue bounds for the preconditioned matrix when the preconditioner is block diagonal and positive definite. We treat linear solves involving the preconditioner as a subproblem which we solve iteratively. In order to minimize cost, we implement a fixed inner tolerance and a varying inner tolerance based on bounds developed by Simoncini and Szyld (2003) and van den Eshof, Sleijpen, and van Gijzen (2005). Numerical experiments compare the cost of preconditioning for various iterative solvers and block preconditioners. We also experiment with different tolerances for the iterative solution of linear solves involving the preconditioner.