Optimal Monte Carlo Updating

TL;DR

This paper derives an optimal transition matrix for Markov chain Monte Carlo based on Peskun's theorem, demonstrating improved efficiency over traditional algorithms through numerical tests on classical and quantum physics models.

Contribution

It introduces a new optimal Monte Carlo updating scheme with zero diagonal elements, enhancing sampling efficiency compared to existing methods.

Findings

Optimal transition matrices have zero diagonal elements except for the largest weight.

Numerical results show improved efficiency over heat-bath and Metropolis algorithms.

Applications include classical Potts model and quantum spin and Bose-Hubbard models.

Abstract

Based on Peskun's theorem it is shown that optimal transition matrices in Markov chain Monte Carlo should have zero diagonal elements except for the diagonal element corresponding to the largest weight. We will compare the statistical efficiency of this sampler to existing algorithms, such as heat-bath updating and the Metropolis algorithm. We provide numerical results for the Potts model as an application in classical physics. As an application in quantum physics we consider the spin 3/2 XY model and the Bose-Hubbard model which have been simulated by the directed loop algorithm in the stochastic series expansion framework.