Deflated BiCGStab for linear equations in QCD problems

TL;DR

This paper explores deflated BiCGStab methods for solving large, complex linear systems in QCD, demonstrating significant improvements by incorporating eigenvector deflation and addressing multiple shifts and right-hand sides.

Contribution

It extends deflation techniques to non-restarted BiCGStab methods for QCD problems, utilizing eigenvectors to accelerate convergence especially near critical parameters.

Findings

Deflating eigenvalues improves BiCGStab performance in difficult QCD problems.

Eigenvector deflation significantly reduces iteration counts.

Projection over previous solutions aids in handling multiple shifts.

Abstract

The large systems of complex linear equations that are generated in QCD problems often have multiple right-hand sides (for multiple sources) and multiple shifts (for multiple masses). Deflated GMRES methods have previously been developed for solving multiple right-hand sides. Eigenvectors are generated during solution of the first right-hand side and used to speed up convergence for the other right-hand sides. Here we discuss deflating non-restarted methods such as BiCGStab. For effective deflation, both left and right eigenvectors are needed. Fortunately, with the Wilson matrix, left eigenvectors can be derived from the right eigenvectors. We demonstrate for difficult problems with kappa near kappa_c that deflating eigenvalues can significantly improve BiCGStab. We also will look at improving solution of twisted mass problems with multiple shifts. Projecting over previous solutions is an easy way to reduce the work needed.