Quasi-Random Sequences and Their Discrepancies

Quasi-random (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the s-dimensional unit cube is measured by its discrepancy, which is of size (log N) N-Ifor large N, as opposed to discrepancy of size (log log N) 1/2 N-1/2 for a random sequence (i.e., for almost any randomly chosen sequence). Several types of discrepancies, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancies are presented for a wide choice of dimension s, number of points N, and different quasi-random sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasi-random sequence is almost exactly the same as that of a randomly chosen sequence. A simplified proof is given for Woniakowski's result relating discrepancy and average integration error, and this result is generalized to other measures on function space. 1. Introduction. Since the beginning of the computer age, Monte Carlo methods have been used to evaluate integrals, solve integral equations, and simulate physical processes [7]. These methods use a sequence ofpoints, usually a deterministic pseudo-random approximation to a randomly chosen sequence, to sample the values of the integrand function or the possible steps in a process. Over the years a number of techniques, such as variance reduction through stratification, have been developed to improve the accuracy of these methods. An alternative technique is to replace the pseudo-random sequence with a deterministic sequence having better uniformity properties. Uniformity of a sequence is measured by its discrepancy, which is the error in representation of the volume of subsets of the unit cube by the fraction of points in the subsets. Several different definitions of discrepancy can be formulated [8], [14], [17], including a sup