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Multilevel frames for sparse tensor product spaces

by R. Harbrecht and R. Schneider and Ch. Schwab

(Report number 2007-06)

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, hypoelliptic 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 [28]. If wavelet bases are not available, we show here how to achieve log-linear complexity 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.

Keywords:

@Techreport{HSS07_367,
  author = {R. Harbrecht and R. Schneider and Ch. Schwab},
  title = {Multilevel frames for sparse tensor product spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2007-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2007/2007-06.pdf },
  year = {2007}
}

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