Improving Multigrid and Conventional Relaxation Algorithms for Propagators

TL;DR

This paper introduces modifications to multigrid and relaxation algorithms that automatically tune parameters, effectively eliminating critical slowing down in propagator computations for bosons and fermions, with demonstrated numerical results.

Contribution

It proposes a new algorithmic modification that removes the need for parameter tuning and prevents critical slowing down in propagator calculations.

Findings

Eliminates critical slowing down in bosonic propagator computations.

Outperforms conjugate gradient in certain volume regimes for bosons.

Numerical results for bosons and staggered fermions in 4D SU(2) gauge fields.

Abstract

Practical modifications of deterministic multigrid and conventional relaxation algorithms are discussed. New parameters need not be tuned but are determined by the algorithms themselves. One modification can be thought of as ``updating on a last layer consisting of a single site''. It eliminates critical slowing down in computations of bosonic and fermionic propagators in a fixed volume. Here critical slowing down means divergence of asymptotic relaxation times as the propagators approach criticality. A remaining volume dependence is weak enough in case of bosons so that conjugate gradient can be outperformed. However, no answer can be given yet if the same is true for staggered fermions on lattices of realizable sizes. Numerical results are presented for propagators of bosons and of staggered fermions in 4-dimensional $SU(2)$ gauge fields.