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

TL;DR

This paper introduces a deflated GMRES-based iterative method for efficiently solving large nonsymmetric linear systems with multiple right-hand sides, especially when small eigenvalues hinder convergence.

Contribution

It presents a novel approach combining deflation with GMRES for multiple right-hand sides, improving efficiency over existing methods and applicable to complex problems like quantum chromodynamics.

Findings

Significant speed-up in solving subsequent systems after initial deflation.

Effective handling of small eigenvalues to accelerate convergence.

Simple implementation compared to other deflation techniques.

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 restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is given showing significant improvement for a problem from quantum chromodynamics. The second and subsequent right-hand sides are solved much quicker than without the deflation. This new approach is relatively simple to implement and is very efficient compared to other deflation methods.