Petras, Knut Smolyak cubature of given polynomial degree with few nodes for increasing dimension. (English) Zbl 1024.65023 Numer. Math. 93, No. 4, 729-753 (2003). This paper deals with the Smolyak algorithm, a procedure that derives numerical cubature methods for tensor product problems from quadrature rules. The number of nodes grow fast for increasing dimensions. It is investigated how to obtain Smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension. A characterization for a subset of Smolyak formulae is given.Error bounds and numerical examples show their behaviour for smooth integrands. A modification is applied to problems of mathematical finance as indicated by a further numerical example. Reviewer: Rudolf Scherer (Karlsruhe) Cited in 18 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A63 Multidimensional problems 91G60 Numerical methods (including Monte Carlo methods) Keywords:Smolyak cubature; cubature formulae; tensor product problems; minimal number of nodes; degree of polynomial exactness; error bounds; numerical examples; mathematical finance Software:PATSYM PDFBibTeX XMLCite \textit{K. Petras}, Numer. Math. 93, No. 4, 729--753 (2003; Zbl 1024.65023) Full Text: DOI