On the Use of Sobol’ Sequence for High Dimensional Simulation

Abstract

When used in simulations, the quasi-Monte Carlo methods utilize specially constructed sequences in order to improve on the respective Monte Carlo methods in terms of accuracy mainly. Their advantage comes from the possibility to devise sequences of numbers that are better distributed in the corresponding high-dimensional unit cube, compared to the randomly sampled points of the typical Monte Carlo method. Perhaps the most widely used family of sequences are the Sobol’ sequences, due to their excellent equidistribution properties. These sequences are determined by sets of so-called direction numbers, where researches have significant freedom to tailor the set being used to the problem at hand. The advancements in scientific computing lead to ever increasing dimensionality of the problems under consideration. Due to the increased computational cost of the simulations, the number of trajectories that can be used is limited. In this work we concentrate on optimising the direction numbers of the Sobol’ sequences in such situations, when the constructive dimension of the algorithm is relatively high, compared to the number of points of the sequence being used. We propose an algorithm that provides us with such sets of numbers, suitable for a range of problems. We then show how the resulting sequences perform in numerical experiments, compared with other well known sets of direction numbers. The algorithm has been efficiently implemented on servers equipped with powerful GPUs and is applicable for a wide range of problems.

This work has been financed in part by a grant from CAF America. We acknowledge the provided access to the e-infrastructure of the Centre for Advanced Computing and Data Processing, with the financial support by the Grant No. BG05M2OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program (2014–2020) and co-financed by the European Union through the European structural and investment funds.