Cost of Generalised HMC Algorithms for Free Field Theory

TL;DR

This paper analytically evaluates the computational efficiency of the Generalised Hybrid Monte Carlo algorithm for free field theory, optimizing parameters and comparing it to other algorithms.

Contribution

It provides an analytical study of GHMC's cost, optimizing parameters, and comparing its efficiency and critical exponents to HMC and L2MC algorithms.

Findings

Long trajectories are optimal for GHMC.

HMC is more efficient than L2MC and Kramers-based algorithms.

HMC and L2MC have the same volume dependence, but different critical exponents.

Abstract

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the autocorrelation functions of operators quadratic in the fields, and optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize. We show that long trajectories are optimal for GHMC, and that standard HMC is much more efficient than algorithms based on the Second Order Langevin (L2MC) or Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z=1 and z=3/2 respectively.