Halftoning and quasi-Monte Carlo

In performing statistical studies in simulation science, it is often necessary to estimate the value of an integral of a function that is based on a simulation calculation. Because simulations tend to be costly in terms of computer time and resources, the ability to accurately estimate integrals with a minimum number of function evaluations is a critical issue in evaluating simulation codes. The goal in Quasi-Monte Carlo (QMC) is to improve the accuracy of integrals estimated by the Monte Carlo technique through a suitable specification of the sample point set. Indeed, the errors from N samples typically drop as N−1 with QMC, which is much better than the N−1/2 dependence obtained with Monte Carlo estimates based on random point sets. The heuristic reasoning behind selecting QMC point sets is similar to that in halftoning, that is, to spread the points out as evenly as possible, consistent with the desired point density. I will outline the parallels between QMC and halftoning, and describe an halftoning-inspired algorithm for generating a sample set with uniform density, which yields smaller integration errors than standard QMC algorithms in two dimensions. An equivalent method, based on repulsive potential fields, is proposed for use in higher dimensions.