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High-Discrepancy Sequences for High-Dimensional Numerical Integration

  • Shu Tezuka
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

In this paper, we consider a sequence of points in [0, 1] d , which are distributed only on the diagonal line between \((0,\ldots ,0)\) and \((1,\ldots ,1)\). The sequence is constructed based on a one-dimensional low-discrepancy sequence. We apply such sequences to d-dimensional numerical integration for two classes of integrals. The first class includes isotropic integrals. Under a certain condition, we prove that the integration error for this class is \(O(\sqrt{\log N}/N)\), where N is the number of points. The second class is called as Kolmogorov superposition integrals for which, under a certain condition, we prove that the integration error for this class is \(O((\log N)/N)\).

Keywords

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Notes

Acknowledgements

The author thanks the anonymous referees for their valuable comments. This research was supported by KAKENHI(22540141).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuoka-shiJapan

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