Optimisations to Hybrid Monte Carlo for Lattice QCD

Abstract

In Lattice Quantum Chromodynamics, we calculate physical quantities on a discrete 4D Euclidean lattice via expectation values, which take the form of path integrals. Due to the high dimensionality of these integrals, the standard technique for evaluating lattice expectation values is Monte Carlo; we generate configurations of gauge fields U and fermion fields distributed according to the lattice action S, then take a weighted average of the observable across the configurations. The most common method used to generate configurations is a Markov Chain technique called Hybrid Monte Carlo. While this technique is functional, it takes a lot of computational resources to generate configurations which are desirably close to the continuum theory. The object of this work is to investigate a variety of improvements over the basic Hybrid Monte Carlo method, and determine which combinations produce independent configurations at the lowest cost. We start by performing a systematic study of filtering for double-flavour simulations, comparing polynomial filtering to the common technique of mass filtering. We show that combining these two methods produces optimal speedup with minimal tuning of parameters, which can be a serious concern when multiple filters are involved. During this investigation, we used the novel technique of overlaid integrators for implementing multiple integration time scales, which expands the possible step-size choices. Next, we investigate improvements to single-flavour simulations, comparing polynomial filtering with a different method that we denote truncated ordered product RHMC. We obtain the best speedup when using truncation filters, but it is highly dependent on the truncation order chosen. To alleviate this problem, we apply a novel integration step-size tuning method called characteristic scale tuning which allows for step-sizes to be better tuned to the energy modes of the system. This improves the performance of our algorithms for a wide range of filter parameters, thus reducing the need to tune filter parameters. Finally, we extend our single-flavour techniques to Lattice QCD+QED simulations, which include electromagnetic effects via a photon field.