Krylov space solvers for shifted linear systems

TL;DR

This paper explores Krylov space methods for efficiently solving multiple shifted linear systems simultaneously, providing a unified framework, analyzing convergence, and demonstrating applications in quantum chromodynamics calculations.

Contribution

It introduces a common framework for shifted Krylov methods, develops short recurrence algorithms like shifted CG and BiCGstab, and applies these to QCD computations.

Findings

Shifted Krylov methods are effective for multiple shifted systems.

The convergence properties of shifted solvers are well understood.

Numerical examples demonstrate applicability in QCD calculations.

Abstract

We investigate the application of Krylov space methods to the solution of shifted linear systems of the form (A+\sigma) x - b = 0 for several values of \sigma simultaneously, using only as many matrix-vector operations as the solution of a single system requires. We find a suitable description of the problem, allowing us to understand known algorithms in a common framework and developing shifted methods basing on short recurrence methods, most notably the CG and the BiCGstab solvers. The convergence properties of these shifted solvers are well understood and the derivation of other shifted solvers is easily possible. The application of these methods to quark propagator calculations in quenched QCD using Wilson and Clover fermions is discussed and numerical examples in this framework are presented. With the shifted CG method an optimal algorithm for staggered fermions is available.