A CG Method for Multiple Right Hand Sides and Multiple Shifts in Lattice QCD Calculations

TL;DR

This paper introduces a new block Krylov subspace method that efficiently solves multiple shifted linear systems with multiple right hand sides, improving computational performance in lattice QCD simulations.

Contribution

A novel block Lanczos-based Krylov method that simultaneously handles multiple shifts and right hand sides, outperforming existing methods.

Findings

Numerical experiments show superior performance over traditional methods.

Method effectively reduces computational time in lattice QCD applications.

Demonstrates advantages in solving systems with multiple shifts and right sides.

Abstract

We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right hand sides. In the past Krylov subspace methods have been developed which exploit either the need for solutions to multiple right hand sides (e.g. deflation type methods and block methods) or multiple shifts (e.g. shifted CG) with some success. In this paper we present a block Krylov subspace method which, based on a block Lanczos process, exploits both features - shifts and multiple right hand sides - at once. Such situations arise, for example, in lattice QCD simulations within the Rational Hybrid Monte Carlo algorithm. We give numerical evidence that our method is superior to applying other iterative methods to each of the systems individually as well as, in some cases, to shifted or block Krylov subspace methods.