PhD thesis of Tim Pillards
Quasi-Monte Carlo integration over a simplex and the entire space
Tim Pilllards
- Advisor: Ronald Cools
- Defense: September 2006
- Local copy: TW2006_05.pdf
Monte Carlo integration is a widely used method to approximate high dimensional integrals. However, the randomness of this method causes the convergence to be very slow. Quasi-Monte Carlo integration uses low discrepancy sequences instead of pseudorandom sequences. The points from these sequences are more uniformly distributed. This causes the convergence to be much faster. Most research in quasi-Monte Carlo concentrates on the unit cube. In this thesis we generalize quasi-Monte Carlo integration to other domains. A large part of this thesis consists of methods to create low discrepancy point sequences in a simplex. We propose several transformations to create low discrepancy sequences in the simplex and compare their performance. We also generalize the Koksma-Hlawka error bound for quasi-Monte Carlo integration on the unit cube to the simplex. Next, we introduce an adaptive quasi-Monte Carlo integration algorithm over the entire space. We prove the optimal distribution of the points for any quasi-Monte Carlo integration over several domains. For quasi-Monte Carlo integration of a function weighted by the normal distribution, the common practice is to transform the integral to the unit cube by the inverse of the cumulative distribution function. We show that the Box-Muller transformation is a worthy alternative. We also present point sets consisting of one or two points that are optimal in function of the discrepancy and investigate the convergence of quasi-Monte Carlo integration of functions of unbounded variation.
