On Figures of Merit for Randomly-Shifted Lattice Rules

Abstract

Randomized quasi-Monte Carlo (RQMC) can be seen as a variance reduction method that provides an unbiased estimator of the integral of a function f over the s-dimensional unit hypercube, with smaller variance than standard Monte Carlo (MC) under certain conditions on f and on the RQMC point set. When f is smooth enough, the variance converges faster, asymptotically, as a function of the number of points n, than the MC rate of $\mathcal{O}(1/n)$. The RQMC point sets are typically constructed to minimize a given parameterized measure of discrepancy between their empirical distribution and the uniform distribution. These parameters can give different weights to the different subsets of coordinates (or lower-dimensional projections) of the points, for example. The ideal parameter values (to minimize the variance) depend very much on the integrand f and their choice (or estimation) is far from obvious in practice. In this paper, we survey this question for randomly-shifted lattice rules, an important class of RQMC point sets, and we explore the practical issues that arise when we want to use the theory to construct lattices for applications. We discuss various ways of selecting figures of merit and for estimating their ideal parameters (including the weights), we examine how they can be implemented in practice, and we compare their performance on examples inspired from real-life problems. In particular, we look at how much improvement (variance reduction) can be obtained, on some examples, by constructing the points based on function-specific figures of merit compared with more traditional general-purpose lattice-rule constructions.