On tractability of path integration

Many applications require approximate values of path integrals. A typical approach is to approximate the path integral by a high dimensional integral and apply a Monte Carlo (randomized) algorithm. However, Monte Carlo algorithm requires roughly e−2 integrand evaluations to provide an e approximation. Moreover, the error bound of e is guaranteed only in a stochastic sense. Do we really need to use randomized algorithms for path integrals? Perhaps, we can find a deterministic algorithm that is more effective even in the worst case setting. To answer this question, we study the worst case complexity of path integration, which, roughly speaking, is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e. We consider path integration with respect to a Gaussian measure, and for various classes of integrands. Tractability of path integration means that the complexity depends polynomially on 1/e. We show that for the class of r times Frechet differentiab...