Numerical integration using sparse grids (Q1281788)

The authors consider various constructions for multivariate quadrature formulas on sparse grids based on Newton-Cotes, Clenshaw-Curtis, Gauss and extended Gauss formulas. They present known results concerning the computational cost and error bounds and indicate a numerically stable implementation. A generalization of \textit{S. A. Smolyak}'s construction [Dokl. Akad. Nauk SSSR 148, 1042-1045 (1963; Zbl 0202.39901)] which can take into account the smoothness properties of the integrand varying with the dimension is given. Using a comparison of various univariate basic integration routines they show that nested quadrature formulas are the best choice for Smolyak's construction. The work contains a bibliography including 56 references from the most significant works in this domain.