Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides

TL;DR

This paper introduces a deflated GMRES method tailored for efficiently solving multiply shifted nonsymmetric linear systems with multiple right-hand sides, significantly reducing computational costs in applications like quantum chromodynamics.

Contribution

It develops a novel deflated GMRES approach that handles multiple shifts and right-hand sides efficiently, improving upon existing methods for complex, non-Hermitian systems.

Findings

Effective in reducing computational costs for multiply shifted systems

Demonstrated success with quantum chromodynamics matrices

Applicable to systems with multiple right-hand sides and shifts

Abstract

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRES-DR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple right-hand sides with multiple shifts for each right-hand side. We develop a method that efficiently solves the multiple right-hand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.