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...

• Type: Journal
• Year: 2007
• Where: JC
• Authors: Michael Gnewuch, René Lindloh, Reinhold Schneider, Anand Srivastav
• Keywords: Certain Multiwavelet Expansions, D-dimensional Unit Cube, JC 2007, Natural D-fold Tensor