On an interpolatory method for high dimensional integration
The numerical integration of functions which are defined on a bounded or unbounded hyperrectangular region is discussed. The integrand must be smooth, possibly very smooth. This means that certain high order derivatives exist and have a small norm. The method is based on a product formula in the sense of \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042-1045 (1963; Zbl 0202.39901)]. Only products with a relatively small number of knots are used. Theoretical results are given concerning error bounds and polynomial exactness. Numerical experiments using the classical formulas of \textit{C. W. Clenshaw} and \textit{A. R. Curtis} [Numer. Math. 2, 197-205 (1960; Zbl 0093.140xx)] and a comparison of different methods are presented. To this end the testing package of Genz functions is used. The paper ends with a report on integrands with Gaussian weight functions, because of their utility in a wide range of applications in physics, statistics, and finance.
scientific article; zbMATH DE number 1398807
scientific article