Parallel Worldline Numerics: Implementation and Error Analysis

Dan Mazur, Jeremy S. Heyl
McGill High Performance Computing Centre, McGill University, 1100 Rue Notre-Dame Ouest, Montreal, QC H3C 1K3
arXiv:1407.7486 [hep-th], (28 Jul 2014)


   title={Parallel Worldline Numerics: Implementation and Error Analysis},

   author={Mazur, Dan and Heyl, Jeremy S.},



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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 xcm, 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.
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