Quasi-Monte Carlo, low discrepancy sequences, and ergodic transformations

Abstract

There has been much recent attention given to the applications of number theory to numerical analysis. One especially noteworthy application involves the numerical approximation of integrals by quasi-Monte Carlo methods (see, for example, Grosswald's 1983 review of a book by Loo Keng Hua and Yuan Wang [3]). It follows from fundamental number theoretic results of Koksma and of Hlawka that good quasi-Monte Carlo formulae evolve by taking nodes from low discrepancy sequences, and so the construction of such sequences becomes important. It will be pointed out how certain low discrepancy sequences may be generated from ergodic transformations. As an instance of this, the well-known van der Corput sequence derives from an example of von Neumann and Kakutani. More specifically, the van der Corput sequence and other low discrepancy sequences can be perceived as orbits of ergodic measure preserving transformations constructed on the unit interval by “splitting and stacking” techniques.