High dimensional integration of smooth functions over cubes

Summary.

We construct a new algorithm for the numerical integration of functions that are defined on a $d$-dimensional cube. It is based on the Clenshaw-Curtis rule for $d=1$ and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and $d \ge 8$.