Introduction

There has been an increasing interest in studying computational aspects of high dimensional problems. Such problems are defined on spaces of functions of d variables and occur in many applications, with d that can be hundreds or even thousands. Examples include:

1. • High dimensional integrals or path integrals with respect to the Wiener measure. These are important for many applications, in particular, in physics and in finance. High dimensional integrals also occur when we want to compute certain parameters of stochastic processes. Moreover, path integrals arise as solutions of partial differential equations given, for example, by the Feynman–Kac formula. See [25, 40, 66, 82, 85, 91].

2. • Global optimization where we need to compute the (global) minimum of a function of d variables. This occurs in many applications, for example, in pattern recognition and in image processing, see [97], or in the modelling and prediction of the geometry of proteins, see [45]. Simulated annealing strategies and genetic algorithms are often used, as well as smoothing techniques and other stochastic algorithms, see [10] and [74]. Some error bounds for deterministic and stochastic algorithms can be found in [42, 43, 44, 48, 53].

3. • The Schrödinger equation for m > 1 particles in ℝ3is a d = 3m-dimensional problem.