Multilevel frames for sparse tensor product spaces

  • Authors:

  • Helmut Harbrecht;Reinhold Schneider;Christoph Schwab

  • Affiliations:

  • Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115, Bonn, Germany;Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623, Berlin, Germany;Eidgenössische Technische Hochschule Zürich, Seminar für Angewandte Mathematik, Rämistr. 101, 8092, Zürich, Switzerland

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2008

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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.