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QMC-integration based on arbitrary (t,m,s)-Nets yields optimal convergence rates for several scales of function spaces

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DREW01 - Multivariate approximation, discretization, and sampling recovery

We study the integration problem over the s-dimensional unit cube on four scales of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter a >0.  We study the worst case error of integration over the norm unit balland provide upper error  bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary (t,m,s)-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary (t,m,s)-nets as sample points yields the best possible rate of convergence.Via suitable embeddings our upper error bounds on Haar wavelet spacestransfer (with possibly different constants) to certain spaces of fractional smoothness 0 < a

This talk is part of the Isaac Newton Institute Seminar Series series.

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