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Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms

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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 provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on weighted Hilbert spaces with an underlying anchored decomposition and arbitrary weights. We focus on randomized algorithms and the randomized worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points. Multilevel algorithms behave very well with respect to the first cost model, while changing dimension algorithms and also dimension-wise quadrature methods, which are based on a similar idea, can take advantage of the more generous second cost model. We prove the first non-trivial lower error bounds for randomized algorithms in these cost models and demonstrate their quality in the case of product weights. In particular, we show that the randomized changing dimension algorithms provided in Plaskota and Wasilkowski (J Complex 27:505–518, 2011) achieve convergence rates arbitrarily close to the optimal convergence rate.

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Acknowledgements

The author gratefully acknowledges support by the German Science Foundation (DFG) under grant GN 91/3-1 and by the Australian Research Council (ARC).

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Authors and Affiliations

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

    Michael Gnewuch

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Michael Gnewuch

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  1. Michael Gnewuch
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Correspondence to Michael Gnewuch .

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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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Gnewuch, M. (2013). Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms. 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_18

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