Mstab: Stabilized Induced Dimension Reduction for Krylov Subspace Recycling

Abstract

We introduce $\mathcal{M}$stab, a Krylov subspace recycling method for the iterative solution of sequences of linear systems, where the system matrix is fixed and is large, sparse, and nonsymmetric, and the right-hand-side vectors are available in sequence. $\mathcal{M}$stab utilizes the short-recurrence principle of induced dimension reduction-type methods, adapted to solve sequences of linear systems. Using IDRstab for solving the linear system with the first right-hand side, the proposed method then recycles the Petrov space constructed throughout the solution of that system, generating a larger initial space for subsequent linear systems. The richer space potentially produces a rapidly convergent scheme. Numerical experiments demonstrate that $\mathcal{M}$stab often enters the superlinear convergence regime faster than other Krylov-type recycling methods.