Numerical integrators for the Hybrid Monte Carlo method

TL;DR

This paper introduces new numerical integrators for Hamiltonian problems used in Hybrid Monte Carlo, focusing on performance with larger step-lengths rather than just accuracy, potentially improving efficiency in high-dimensional sampling.

Contribution

It proposes a novel methodology for constructing integrators based on sampling performance, leading to new splitting formulae requiring multiple force evaluations per step.

Findings

New integrators show potential efficiency improvements over Verlet

Performance benefits are notable in high-dimensional problems

Limited experiments suggest practical advantages

Abstract

We construct numerical integrators for Hamiltonian problems that may advantageously replace the standard Verlet time-stepper within Hybrid Monte Carlo and related simulations. Past attempts have often aimed at boosting the order of accuracy of the integrator and/or reducing the size of its error constants; order and error constant are relevant concepts in the limit of vanishing step-length. We propose an alternative methodology based on the performance of the integrator when sampling from Gaussian distributions with not necessarily small step-lengths. We construct new splitting formulae that require two, three or four force evaluations per time-step. Limited, proof-of-concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard Verlet method, especially in problems with high dimensionality.