Optimized general sparse grid approximation spaces for operator equations
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Abstract:
This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, we construct operator-adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces, provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, their dimension is $O(2^{J})$ independent of the dimension $n$ of the problem, compared to $O(2^{Jn})$ for the full grid space. Also, for operators of negative order the overall cost is significantly in favor of the new approximation spaces. We give cost estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a Galerkin method together with optimal preconditioning. The theory covers elliptic boundary value problems as well as boundary integral equations.References
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Additional Information
- M. Griebel
- Affiliation: Institut für Numerische Simulation, Universität Bonn, D-53115 Bonn, Germany
- MR Author ID: 270664
- Email: griebel@ins.uni-bonn.de
- S. Knapek
- Affiliation: Institut für Numerische Simulation, Universität Bonn, D-53115 Bonn, Germany
- Received by editor(s): April 10, 2008
- Received by editor(s) in revised form: December 4, 2008
- Published electronically: April 23, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 2223-2257
- MSC (2000): Primary 41A17, 41A25, 41A30, 41A65, 45L10, 65D99, 65N12, 65N30, 65N38, 65N55
- DOI: https://doi.org/10.1090/S0025-5718-09-02248-0
- MathSciNet review: 2521287