Quasi-Monte Carlo methods for high dimensional integration - the standard (weighted Hilbert space) setting and beyond

Abstract

This paper is a contemporary review of QMC ("quasi-Monte Carlo") methods, i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube $[0; 1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast CBC ("component-by-component") construction of lattice rules that achieve the optimal convergence order (i.e., a rate of almost $1=N$, where $N$ is the number of points, independently of dimension) to so-called POD ("product-and-order-dependent") weights, as seen in some recent applications.