Quasi-Monte Carlo Integration on the Unit Disk

Quasi-Monte Carlo integration uses deterministic, low discrepancy sequences, such as the Halton or the Sobol sequence, to carry out numerical integration or search. These quasi random sequences are defined over the unit interval, or its multidimensional analogues, i.e. the unit square, cube, or hypercube. However, there are applications where we need quasi random sequences that are defined over a circular, spherical, or ellipsoidal domains. We study the performance of quasi random sequences for integration on a unit disk by generating quasi-random sequences defined over the unit square and transforming them to sequences defined on the unit disk by using coordinate transformations. We evaluate integrals defined over a circular disk and find that quasi-Monte Carlo integration using Sobol sequences outperforms Monte Carlo integration using pseudo-random sequences.