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Infinite-Dimensional Quadrature and Approximation of Distributions

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Abstract

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Jakob Creutzig & Klaus Ritter

  2. Institut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Steffen Dereich

  3. Institut für Mathematische Stochastik, Fakultät für Mathematik, Universität Magdeburg, Postfach 4120, 39016, Magdeburg, Germany

    Thomas Müller-Gronbach

Authors
  1. Jakob Creutzig
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  2. Steffen Dereich
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  3. Thomas Müller-Gronbach
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  4. Klaus Ritter
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Correspondence to Steffen Dereich.

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Communicated by Arieh Iserles.

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Creutzig, J., Dereich, S., Müller-Gronbach, T. et al. Infinite-Dimensional Quadrature and Approximation of Distributions. Found Comput Math 9, 391–429 (2009). https://doi.org/10.1007/s10208-008-9029-x

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