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Adaptive Wavelet Methods for SPDEs

  • Petru A. Cioica
  • Stephan Dahlke
  • Nicolas Döhring
  • Stefan Kinzel
  • Felix Lindner
  • Thorsten Raasch
  • Klaus Ritter
  • René L. SchillingEmail author
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (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 = sd + 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.

Keywords

Besov Space Wavelet Basis Weighted Sobolev Space Adaptive Wavelet Bound Lipschitz Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Petru A. Cioica
    • 4
  • Stephan Dahlke
    • 4
  • Nicolas Döhring
    • 2
  • Stefan Kinzel
    • 4
  • Felix Lindner
    • 2
  • Thorsten Raasch
    • 3
  • Klaus Ritter
    • 2
  • René L. Schilling
    • 1
    Email author
  1. 1.Technical University of DresdenDresdenGermany
  2. 2.Technical University of KaiserslauternKaiserslauternGermany
  3. 3.University of MainzMainzGermany
  4. 4.Philipps-University of MarburgMarburgGermany

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