Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

TL;DR

This paper extends the recycling BiCGSTAB algorithm to efficiently solve sequences of non-symmetric linear systems, demonstrating significant computational savings in parametric model order reduction applications.

Contribution

The paper generalizes recycling BiCGSTAB for non-symmetric systems, incorporating invariant subspaces to improve efficiency in parametric model order reduction.

Findings

40% reduction in matrix-vector products

35% reduction in runtime

Effective for parametric model order reduction

Abstract

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.