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On the component by component construction of polynomial lattice point sets for numerical integration in weighted Sobolev spaces

Author(s): Peter Kritzer | Friedrich Pillichshammer

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 6;
Issue: 1;
Start page: 79;
Date: 2011;
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Keywords: Quasi-Monte Carlo | polynomial lattice rules | weighted Sobolev spaces | Hilbert space with kernel | Walsh function

Polynomial lattice point sets are polynomial versions of classical lattice point sets and among the most widely used classes of node sets for quasi-Monte Carlo integration. In this paper, we study the worst-case integration error of digitally shifted polynomial lattice point sets and give step by step construction algorithms to obtain polynomial lattices that achieve a low worst-case error in certain weighted Sobolev spaces. The construction algorithm is a so-called component by component algorithm, choosing one component of the relevant point set at a time. Furthermore, under certain conditions on the weights, we achieve that there is only a polynomial or even no dependence of the worst-case error on the dimension of the integration problem.

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