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# Multilevel frames for sparse tensor product spaces

## Harbrecht, Helmut ; Schneider, Reinhold ; Schwab, Christoph

Numerische Mathematik, 2008, Vol.110(2), pp.199-220 [Peer Reviewed Journal]

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• Title:
Multilevel frames for sparse tensor product spaces
• Author/Creator: Harbrecht, Helmut ; Schneider, Reinhold ; Schwab, Christoph
• Language: English
• Subjects: 35J25 ; 35R60 ; 65N30 ; 65F50
• Is Part Of: Numerische Mathematik, 2008, Vol.110(2), pp.199-220
• Description: 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.
• Identifier: ISSN: 0029-599X ; E-ISSN: 0945-3245 ; DOI: 10.1007/s00211-008-0162-x