Testing multidimensional integration routines
Proc. of international conference on Tools, methods and languages for scientific and engineering computation
Information-based complexity
A class of bases in L2 for the sparse representations of integral operators
SIAM Journal on Mathematical Analysis
Explicit cost bounds of algorithms for multivariate tensor product problems
Journal of Complexity
Monte Carlo Variance of Scrambled Net Quadrature
SIAM Journal on Numerical Analysis
Multiwavelets for Second-Kind Integral Equations
SIAM Journal on Numerical Analysis
Optimal quadrature for Haar wavelet spaces
Mathematics of Computation
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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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 H^s[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 methods.