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Numerical integration using sparse grids. (English) Zbl 0921.65022
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 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.
Reviewer: D.Acu (Sibiu)

MSC:
65D32 Numerical quadrature and cubature formulas
65Y20 Complexity and performance of numerical algorithms
41A55 Approximate quadratures
41A63 Multidimensional problems
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