HIGH-DIMENSIONAL INTEGRATION OF SMOOTH FUNCTIONS OVER CUBES

We construct a new algorithm for the numerical integration of function s that are defined on a d-dimensional cube. It is based on the Clensha w-Curtis rule for d = 1 and on Smolyak's construction. This way we mak e the best use of the smoothness properties of any (nonperiodic) funct ion. We prove error bounds showing that our algorithm is almost optima l (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method i s very competitive, in particular for smooth integrands and d greater than or equal to 8.