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Cubature formulas for function spaces with moderate smoothness

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Cubature formulas for function spaces with moderate smoothness
We construct simple algorithms for high-dimensional numerical integration of function classes with moderate smoothness. These classes consist of square-integrable functions over the d-dimensional unit cube whose coefficients with respect to certain multiwavelet expansions decay rapidly. Such a class contains discontinuous functions on the one hand and, for the right choice of parameters, the quite natural d-fold tensor product of a Sobolev space Hs[0, 1] on the other hand. The algorithms are based on one-dimensional quadrature rules appropriate for the integration of the particular wavelets under consideration and on Smolyak’s construction. We provide upper bounds for the worst-case error of our cubature rule in terms of the number of function calls. We additionally prove lower bounds showing that our method is optimal in dimension d = 1 and almost optimal (up to logarithmic factors) in higher dimensions. We perform numerical tests which allow the comparison with other cubature meth...
Michael Gnewuch, René Lindloh, Reinhold Sch
Added 15 Dec 2010
Updated 15 Dec 2010
Type Journal
Year 2007
Where JC
Authors Michael Gnewuch, René Lindloh, Reinhold Schneider, Anand Srivastav
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