High-dimensional numerical integration

Abstract

Multiple integrals of high dimensions, often running into the hundreds or even thousands, occur in various applications, notably in computational finance. Classical methods for approximating multiple integrals based on Cartesian products of one-dimensional integration rules (trapezoidal rule, Simpson's rule, etc.) are efficient only in low-dimensional cases. For many years the method of choice for high-dimensional numerical integration was the statistical Monte Carlo method. However, the deterministic version of the Monte Carlo method, called the quasi-Monte Carlo method, offers a considerably faster convergence rate and is thus gradually replacing the statistical Monte Carlo method in many applications. In this talk we discuss the above methods, with the emphasis being on the quasi-Monte Carlo method and its applications. A fascinating problem is that of the explicit construction of suitable node sets for quasi-Monte Carlo integration. Surprising connections with areas of discrete mathematics, such as coding theory, arise in this context.