Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides

TL;DR

This paper introduces deflated iterative methods for efficiently solving large nonsymmetric linear systems with multiple right-hand sides, leveraging eigenvector information to accelerate convergence, demonstrated with applications in quantum chromodynamics.

Contribution

It presents a novel approach combining deflated GMRES and projection techniques for multiple right-hand sides, improving efficiency over traditional methods.

Findings

Significant convergence improvements shown in quantum chromodynamics example

Effective combination of deflation with restarted and non-restarted methods

Alternative to block methods with comparable or better performance

Abstract

A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple right-hand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining an iterative method with a projection over the previously determined eigenvectors. Restarted GMRES is considered for the iterative method as well as non-restarted methods such as BiCGSTAB. These methods offer an alternative to block methods, and they can also be combined with a block approach. An example is given showing significant improvement for a problem from quantum chromodynamics.