Efficient Preconditioning of Sequences of Nonsymmetric Linear Systems

We present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems. This approach is algebraic and it is theoretically motivated. It generalizes diagonal updates introduced by Benzi and Bertaccini [BIT, 43 (2003), pp. 231–244] and Bertaccini [Electron. Trans. Numer. Anal., 18 (2004), pp. 49–64]. It is shown experimentally that this approach can be very beneficial. For example, it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence. In some cases, the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners. Since the updates are typically cheap and straightforward, their use is of practical interest. They can replace recomputing preconditioners, which is often expensive, especially in parallel and matrix-free environments.