A SQMRCGstab Algorithm for Families of Shifted Linear Systems

TL;DR

This paper introduces the SQMRCGstab algorithm, an efficient iterative method for solving families of shifted linear systems, particularly from Quantum Chromodynamics, improving convergence and computational cost over existing methods.

Contribution

The paper develops a novel shifted QMRCGstab method that extends quasi-minimum residual techniques to shifted systems, enhancing efficiency and residual smoothing.

Findings

The SQMRCGstab method reduces matrix-vector products and inner products per system.

It achieves better residual smoothing compared to shifted BiCGstab.

Numerical tests demonstrate superior efficiency on QCD problems.

Abstract

This study is mainly focused on iterative solutions to shifted linear systems arising from a Quantum Chromodynamics (QCD) problem. To solve such system efficiently, we explore a kind of shifted QMRCGstab (SQMRCGstab) methods, which is derived by extending the quasi-minimum residual to the shifted BiCGstab. The shifted QMRCGstab method takes advantage of the shifted structure, so that the number of matrix-vector products and the number of inner products are the same as a single linear system. Moreover, the SQMRCGstab achieves a smoothing of the residual compared to the shifted BiCGstab, and is more competitive than the MS-QMRIDR(s) and the shifted BiCGstab on the QCD problem. Numerical examples show also the efficiency of the method when one applies it to the real problems.