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Adaptive Wavelet Methods for SPDEs

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Extraction of Quantifiable Information from Complex Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

Abstract

We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains \(\mathcal{O}\subset \mathbb{R}^{d}\). A detailed regularity analysis for the solution process u in the scale of Besov spaces \(B_{\tau,\tau }^{s}(\mathcal{O})\), 1∕τ = s∕d + 1∕p, α > 0, p ≥ 2, is presented. The regularity in this scale is known to determine the order of convergence that can be achieved by adaptive wavelet algorithms and other nonlinear approximation schemes. As it turns out, in general, for solutions of SPDEs this regularity exceeds the \(L_{p}(\mathcal{O})\)-Sobolev regularity, which determines the order of convergence for uniform approximation schemes. We also study nonlinear wavelet approximation of elliptic boundary value problems on \(\mathcal{O}\) with random right-hand side. Such problems appear naturally when applying Rothe’s method to the parabolic stochastic equation. A general stochastic wavelet model for the right-hand side is introduced and its Besov regularity as well as linear and nonlinear approximation is studied. The results are matched by computational experiments.

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Author information

Authors and Affiliations

  1. Technical University of Dresden, Zellescher Weg 12–14, 01069, Dresden, Germany

    René L. Schilling

  2. Technical University of Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Nicolas Döhring, Felix Lindner & Klaus Ritter

  3. University of Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Thorsten Raasch

  4. Philipps-University of Marburg, Hans Meerwein Str., Lahnberge, 35032, Marburg, Germany

    Petru A. Cioica, Stephan Dahlke & Stefan Kinzel

Authors
  1. Petru A. Cioica
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  2. Stephan Dahlke
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  3. Nicolas Döhring
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  4. Stefan Kinzel
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  5. Felix Lindner
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  6. Thorsten Raasch
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  7. Klaus Ritter
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  8. René L. Schilling
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Corresponding author

Correspondence to René L. Schilling .

Editor information

Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

  3. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

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

    Klaus Ritter

  6. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

  8. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Harry Yserentant

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Cioica, P.A. et al. (2014). Adaptive Wavelet Methods for SPDEs. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_5

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