Weighted integration over a cube based on digital nets and sequences

TL;DR

This paper extends Quasi-Monte Carlo integration to general product measures on arbitrary cubes using digital nets and sequences, providing error bounds independent of density smoothness.

Contribution

It introduces a method for QMC integration over arbitrary cubes with general product measures, including error bounds and construction techniques that are measure-independent.

Findings

Error bounds depend only on the number of points and dimension.

The method applies to non-smooth density functions.

Component-by-component construction is measure-independent.

Abstract

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the cube but is otherwise independent of the product measure.