High-dimensional integration: The quasi-Monte Carlo way

TLDR

Quasi‑Monte Carlo (QMC) methods provide equal‑weight rules for approximating high‑dimensional integrals over the unit cube, and this review surveys their contemporary development. The paper surveys recent advances in lattice methods, digital nets, and related constructions aimed at producing QMC rules with specified convergence rates and controlled worst‑case error growth as dimension increases. Key to these constructions are weight parameters and weighted function spaces, with analysis relying on reproducing kernel Hilbert spaces, discrepancy, and related tools.

Abstract

This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.