Parallel Worldline Numerics: Implementation and Error Analysis

TL;DR

This paper discusses the implementation of parallel worldline numerics using CUDA, focusing on calculating Wilson loops in magnetic fields, and analyzes the associated uncertainties with a novel jackknife method.

Contribution

It introduces a GPU-based parallel implementation of worldline numerics and provides an in-depth analysis of statistical uncertainties and error estimation techniques.

Findings

GPU implementation achieves significant speedup

Uncertainty analysis accounts for non-Gaussian distributions

Jackknife method effectively estimates correlated uncertainties

Abstract

We give an overview of the worldline numerics technique, and discuss the parallel CUDA implementation of a worldline numerics algorithm. In the worldline numerics technique, we wish to generate an ensemble of representative closed-loop particle trajectories, and use these to compute an approximate average value for Wilson loops. We show how this can be done with a specific emphasis on cylindrically symmetric magnetic fields. The fine-grained, massive parallelism provided by the GPU architecture results in considerable speedup in computing Wilson loop averages. Furthermore, we give a brief overview of uncertainty analysis in the worldline numerics method. There are uncertainties from discretizing each loop, and from using a statistical ensemble of representative loops. The former can be minimized so that the latter dominates. However, determining the statistical uncertainties is complicated by two subtleties. Firstly, the distributions generated by the worldline ensembles are highly non-Gaussian, and so the standard error in the mean is not a good measure of the statistical uncertainty. Secondly, because the same ensemble of worldlines is used to compute the Wilson loops at different values of $T$ and $x_\mathrm{ cm}$, the uncertainties associated with each computed value of the integrand are strongly correlated. We recommend a form of jackknife analysis which deals with both of these problems.