We will present methods for improving the performance and robustness of incomplete factorization preconditioners for solving both SPD and general sparse linear systems. A key technique that results in signiﬁcant improvement is the use of block rows and columns, which is commonplace in direct solvers. This enables the construction of denser and more robust preconditioners without the excessive cost traditionally associated with the construction of such preconditioners. We present empirical results to show that in most cases, the preconditioner density can be increased in a memory-neutral manner for solving unsymmetric systems with restarted GMRES because the restart parameter can be reduced without compromising robustness. In addition to blocking, we will also present some relatively simple but effective adaptive heuristics for improving preconditioning based on incomplete factorization. These include robust management of breakdown of incomplete Cholesky factorization, use of a ﬂexible ﬁll-factor, and automatic tuning of drop tolerance and ﬁll-factor in applications requiring solution of a sequence of linear systems.
Business Analytics and Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, US