Optimization of the Direction Numbers of the Sobol Sequences

摘要

The quasi-Monte Carlo methods use specially crafted sequences instead of the usual pseudo-random sequences used in Monte Carlo methods, in order to improve on the rate of convergence. Most of the time these sequences are of low-discrepancy, with the Sobol sequences being one of the most widely used in practice, especially in Financial Mathematics. The definition of the Sobol sequences offers substantial freedom in choosing the so-called direction numbers, so that the basic properties of the sequences are ensured, while additional equi-distribution properties are achieved through some optimization procedure. In a particular computation when the number of dimensions is relatively high compared with the number of terms of the sequence,pertain artefacts in the distribution of the sequences become difficult to avoid. In this work we describe our approach on how to define additional equi-distribution criteria and the optimization procedure that we used in order to obtain new sets of direction numbers usable in such settings. Numerical results demonstrating the performance of the sequences using these direction numbers are also discussed.