Coarsest-level improvements in multigrid for lattice QCD on large-scale computers

TL;DR

This paper explores novel enhancements to the coarsest-level solver in multigrid methods for lattice QCD, significantly improving scalability and robustness for large-scale quantum chromodynamics simulations.

Contribution

It introduces and evaluates combined deflation and polynomial preconditioning techniques to enhance the coarsest-level solver in algebraic multigrid methods for lattice QCD.

Findings

Deflation and polynomial preconditioning improve solver performance at small mass parameters.

Enhanced insensitivity to conditioning in the Wilson case.

Eliminated artificial parameter adjustments in twisted mass discretization.

Abstract

Numerical simulations of quantum chromodynamics (QCD) on a lattice require the frequent solution of linear systems of equations with large, sparse and typically ill-conditioned matrices. Algebraic multigrid methods are meanwhile the standard for these difficult solves. Although the linear systems at the coarsest level of the multigrid hierarchy are much smaller than the ones at the finest level, they can be severely ill-conditioned, thus affecting the scalability of the whole solver. In this paper, we investigate different novel ways to enhance the coarsest-level solver and demonstrate their potential using DD-$\alpha$AMG, one of the publicly available algebraic multigrid solvers for lattice QCD. We do this for two lattice discretizations, namely clover-improved Wilson and twisted mass. For both the combination of two of the investigated enhancements, deflation and polynomial preconditioning, yield significant improvements in the regime of small mass parameters. In the clover-improved Wilson case we observe a significantly improved insensitivity of the solver to conditioning, and for twisted mass we are able to get rid of a somewhat artificial increase of the twisted mass parameter on the coarsest level used so far to make the coarsest level solves converge more rapidly.