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Adaptive wavelet methods for the stochastic Poisson equation

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Abstract

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝd. The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.

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Acknowledgements

We are grateful to Natalia Bochkina for valuable discussions, and we thank Tiange Xu for her contributions at an early stage of the project. Also, we thank Winfried Sickel for his comments on regularity theory.

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Authors and Affiliations

  1. Philipps-Universität Marburg, Marburg, Germany

    Petru A. Cioica, Stephan Dahlke & Stefan Kinzel

  2. TU Dresden, Dresden, Germany

    Felix Lindner & René L. Schilling

  3. Johannes Gutenberg-Universität Mainz, Mainz, Germany

    Thorsten Raasch

  4. TU Kaiserslautern, Kaiserslautern, Germany

    Nicolas Döhring & Klaus Ritter

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 Stefan Kinzel.

Additional information

Communicated by Desmond Higham.

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG, grants DA 360/13-1, RI 599/4-1, SCHI 419/5-1) and a doctoral scholarship of the Philipps-Universität Marburg.

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Cioica, P.A., Dahlke, S., Döhring, N. et al. Adaptive wavelet methods for the stochastic Poisson equation. Bit Numer Math 52, 589–614 (2012). https://doi.org/10.1007/s10543-011-0368-7

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