Numerical implementation of some reweighted path integral methods

TL;DR

This paper investigates two reweighted path integral methods based on Levy-Ciesielski and Wiener-Fourier series, demonstrating their cubic convergence for smooth potentials and proposing quadrature techniques to preserve this convergence.

Contribution

It introduces and analyzes two specific reweighted path integral methods, providing numerical evidence of their convergence rates and proposing quadrature techniques to maintain accuracy.

Findings

Cubic convergence for smooth potentials.

Numerical validation of theoretical predictions.

Quadrature techniques that preserve asymptotic convergence.

Abstract

The reweighted random series techniques provide finite-dimensional approximations to the quantum density matrix of a physical system that have fast asymptotic convergence. We study two special reweighted techniques that are based upon the Levy-Ciesielski and Wiener-Fourier series, respectively. In agreement with the theoretical predictions, we demonstrate by numerical examples that the asymptotic convergence of the two reweighted methods is cubic for smooth enough potentials. For each reweighted technique, we propose some minimalist quadrature techniques for the computation of the path averages. These quadrature techniques are designed to preserve the asymptotic convergence of the original methods.