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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 (should also be assigned at least one other classification number from Section 41-XX)
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