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Hit-and-Run for Numerical Integration

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Monte Carlo and Quasi-Monte Carlo Methods 2012

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 65))

Abstract

We study the numerical computation of an expectation of a bounded function f with respect to a measure given by a non-normalized density on a convex body \(K \subset {\mathbb{R}}^{d}\). We assume that the density is log-concave, satisfies a variability condition and is not too narrow. In [19, 25, 26] it is required that K is the Euclidean unit ball. We consider general convex bodies or even the whole \({\mathbb{R}}^{d}\) and show that the integration problem satisfies a refined form of tractability. The main tools are the hit-and-run algorithm and an error bound of a multi run Markov chain Monte Carlo method.

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Acknowledgements

The author gratefully acknowledges the comments of the referees and wants to express his thanks to the local organizers of the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing for their hospitality. The research was supported by the DFG Priority Program 1324, the DFG Research Training Group 1523, and an Australian Research Council Discovery Project.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Friedrich-Schiller-University Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Daniel Rudolf

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia

    Daniel Rudolf

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  1. Daniel Rudolf
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Correspondence to Daniel Rudolf .

Editor information

Editors and Affiliations

  1. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Josef Dick

  2. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Frances Y. Kuo

  3. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Gareth W. Peters

  4. The University of New South Wales School of Mathematics and Statistics, Sydney, New South Wales, Australia

    Ian H. Sloan

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Rudolf, D. (2013). Hit-and-Run for Numerical Integration. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_31

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