Krylov Subspace Recycling for Sequences of Shifted Linear Systems

TL;DR

This paper investigates Krylov subspace recycling methods for efficiently solving sequences of shifted linear systems, proposing two schemes to handle the challenges of shared augmented subspaces and demonstrating their effectiveness with applications in quantum chromodynamics.

Contribution

It introduces two novel schemes for Krylov subspace recycling tailored for shifted linear systems, addressing storage and efficiency limitations of existing methods.

Findings

One scheme constructs separate deflation spaces for each shifted system.

The other builds a single recycled subspace and iteratively refines solutions.

Numerical examples show effectiveness in lattice quantum chromodynamics applications.

Abstract

We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.