Smolyak cubature of given polynomial degree with few nodes for increasing dimension
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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.
minimal number of nodes, mathematical finance, Smolyak cubature, tensor product problems, numerical examples, Numerical methods (including Monte Carlo methods), degree of polynomial exactness, Multidimensional problems, error bounds, Numerical quadrature and cubature formulas, cubature formulae
minimal number of nodes, mathematical finance, Smolyak cubature, tensor product problems, numerical examples, Numerical methods (including Monte Carlo methods), degree of polynomial exactness, Multidimensional problems, error bounds, Numerical quadrature and cubature formulas, cubature formulae
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