Many Masses on One Stroke: Economic Computation of Quark Propagators

TL;DR

This paper introduces efficient algorithms for computing multiple quark propagators simultaneously in lattice QCD by exploiting matrix structures, significantly reducing computational effort especially for small quark masses.

Contribution

It presents a novel approach using QMR and BCG methods to compute entire trajectories of quark masses at once, improving efficiency over traditional methods.

Findings

QMR and BCG methods outperform standard algorithms in the critical small-mass regime.

Simultaneous computation of multiple propagators reduces overall computational cost.

Using symmetry properties cuts computational effort by a factor of two.

Abstract

The computational effort in the calculation of Wilson fermion quark propagators in Lattice Quantum Chromodynamics can be considerably reduced by exploiting the Wilson fermion matrix structure in inversion algorithms based on the non-symmetric Lanczos process. We consider two such methods: QMR (quasi minimal residual) and BCG (biconjugate gradients). Based on the decomposition $M/\kappa={\bf 1}/\kappa-D$ of the Wilson mass matrix, using QMR, one can carry out inversions on a {\em whole} trajectory of masses simultaneously, merely at the computational expense of a single propagator computation. In other words, one has to compute the propagator corresponding to the lightest mass only, while all the heavier masses are given for free, at the price of extra storage. Moreover, the symmetry $\gamma_5\, M= M^{\dagger}\,\gamma_5$ can be used to cut the computational effort in QMR and BCG by a factor of two. We show that both methods then become---in the critical regime of small quark masses---competitive to BiCGStab and significantly better than the standard MR method, with optimal relaxation factor, and CG as applied to the normal equations.