Many Masses on One Stroke: Economic Computation of Quark Propagators

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 $M/κ={\bf 1}/κ-D$ 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 $γ_5\, M= M^{\dagger}\,γ_5$ 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.