What some people do to their "high-dimensional" integrals

This talk gives an introduction to quasi-Monte Carlo methods for high-dimensional integrals. Such methods apply low-discrepancy points in an equal-weights quadrature rule, in contrast to the Monte Carlo method which uses random points. Low-discrepancy point sets are deterministic point sets dependent on some parameters and with a specific structure. We first start by motivating the usage of Monte Carlo and then quasi-Monte Carlo after which we then explore some of the recent developments. These topics include: worst-case errors in reproducing kernel Hilbert spaces, construction of lattice rules/sequences, the effective dimension, and higher order of convergence. In the minds of many, quasi-Monte Carlo methods seem to share the bad stanza of the Monte Carlo method: a brute force method of last resort with slow order of convergence, i.e., $O(N^{-1/2})$. This is not so. While the standard rate of convergence for quasi-Monte Carlo is rather slow, being $O(N^{-1})$, the theory shows that these methods achieve the optimal rate of convergence in many interesting function spaces. E.g., in function spaces with higher smoothness one can have $O(N^{-\alpha})$, $\alpha > 1$. This will be illustrated by numerical examples. This talk is meant for a general audience.status: publishe