On the efficient Monte Carlo implementation of path integrals

TL;DR

This paper presents an efficient Monte Carlo method for path integrals using Levy-Ciesielski representation, enabling faster computation and sampling, with flexibility in the number of time slices for different degrees of freedom, improving simulation efficiency.

Contribution

The paper introduces a Levy-Ciesielski based implementation for path integrals that enhances computational efficiency and sampling speed in Monte Carlo simulations, with layered path variables for independence.

Findings

Sequential sampling matches or exceeds all-particle updates in efficiency.

Layered path variables are statistically independent, enabling faster sampling.

Numerical simulations confirm computational advantages of the proposed method.

Abstract

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.