Deflation and Flexible SAP-Preconditioning of GMRES in Lattice QCD Simulation

TL;DR

This paper introduces a novel combination of deflation techniques and flexible SAP-preconditioning within GMRES to improve the convergence of lattice QCD simulations, especially for poorly conditioned systems.

Contribution

It develops a new hybrid algorithm integrating SAP and FGMRES-DR, enhancing convergence in lattice QCD linear system solutions with poor conditioning.

Findings

FGMRES alone stagnates on poorly conditioned systems.

Adding deflation via FGMRES-DR improves convergence.

The combined method effectively solves challenging lattice QCD problems.

Abstract

The simulation of lattice QCD on massively parallel computers stimulated the development of scalable algorithms for the solution of sparse linear systems. We tackle the problem of the Wilson-Dirac operator inversion by combining a Schwarz alternating procedure (SAP) in multiplicative form with a flexible variant of the GMRES-DR algorithm. We show that restarted GMRES is not able to converge when the system is poorly conditioned. By adding deflation in the form of the FGMRES-DR algorithm, an important fraction of the information produced by the iterates is kept between successive restarts leading to convergence in cases in which FGMRES stagnates.