Average case complexity of multivariate integration

Abstract

We study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical Wiener sheet measure. To derive the average case complexity one needs to obtain optimal sample points. This optimal design problem has long been open. All known designs guaranteeing average case error ε lead to an exponential number of sample points, roughly Θ (ε‒d). For d large this makes the problem intractable for even the fastest computers. Yet good designs have to exist since the average case com-plexity is bounded by O(ε‒2) as can be proven by considering the Monte-Carlo algorithm. We just did not know how to construct them. In this paper we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years. of particular importance for our purpose are papers by Roth [10, 11]. This relation enables us to show that optimal sample points can be derived from Hammersley points. Extending the result of Roth [10] and using the recent result of Wasilkowski [19], we conclude that the average case complexity is Θ(ε ‒1(ln ε‒l)(d‒1)/2). © 1991 American Mathematical Society.