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Monte Carlo and Quasi-Monte Carlo Methods 2012 pp 297–316Cite as

Complexity of Banach Space Valued and Parametric Integration

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

Abstract

We study the complexity of Banach space valued integration. The input data are assumed to be r-smooth. We consider both definite and indefinite integration and analyse the deterministic and the randomized setting. We develop algorithms, estimate their error, and prove lower bounds. In the randomized setting the optimal convergence rate turns out to be related to the geometry of the underlying Banach space. Then we study the corresponding problems for parameter dependent scalar integration. For this purpose we use the Banach space results and develop a multilevel scheme which connects Banach space and parametric case.

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Author information

Authors and Affiliations

  1. University of Kaiserslautern, D-67653, Kaiserslautern, Germany

    Thomas Daun & Stefan Heinrich

Authors
  1. Thomas Daun
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  2. Stefan Heinrich
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Corresponding author

Correspondence to Thomas Daun .

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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Daun, T., Heinrich, S. (2013). Complexity of Banach Space Valued and Parametric 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_12

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