Deflated GMRES with Multigrid for Lattice QCD

TL;DR

This paper investigates multigrid deflation techniques for lattice QCD solvers, demonstrating improved scaling and efficiency at coarse grid levels, especially near critical parameters, using GMRES-DR and GMRES-Proj algorithms.

Contribution

It introduces a multigrid deflation approach on the coarse grid for lattice QCD, showing significant scaling improvements over traditional methods.

Findings

Deflation at the coarse grid improves solver scaling.

Partial solves on intermediate grids enhance coarse grid deflation.

GMRES-DR and GMRES-Proj algorithms perform well with coarse grid deflation.

Abstract

Lattice QCD solvers encounter critical slowing down for fine lattice spacings and small quark mass. Traditional matrix eigenvalue deflation is one approach to mitigating this problem. However, to improve scaling we study the effects of deflating on the coarse grid in a hierarchy of three grids for adaptive mutigrid applications of the two dimensional Schwinger model. We compare deflation at the fine and coarse levels with other non deflated methods. We find the inclusion of a partial solve on the intermediate grid allows for a low tolerance deflated solve on the coarse grid. We find very good scaling in lattice size near critical mass when we deflate at the coarse level using the GMRES-DR and GMRES-Proj algorithms.