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Numerische Mathematik

, 110:199 | Cite as

Multilevel frames for sparse tensor product spaces

  • Helmut HarbrechtEmail author
  • Reinhold Schneider
  • Christoph Schwab
Article

Abstract

For Au = f with an elliptic differential operator \({A:\mathcal{H} \rightarrow \mathcal{H}'}\) and stochastic data f, the m-point correlation function \({{\mathcal M}^m u}\) of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A (m) of A. Sparse tensor products of hierarchic FE-spaces in \({\mathcal{H}}\) are known to allow for approximations to \({{\mathcal M}^m u}\) which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of \({{\mathcal M}^m u}\) for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.

Mathematics Subject Classification (2000)

35J25 35R60 65N30 65F50 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Helmut Harbrecht
    • 1
    Email author
  • Reinhold Schneider
    • 2
  • Christoph Schwab
    • 3
  1. 1.Institut für Numerische SimulationUniversität BonnBonnGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Seminar für Angewandte MathematikEidgenössische Technische Hochschule ZürichZürichSwitzerland

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