DagSemProc.04401.12.pdf
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Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings
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Dagstuhl Seminar Proceedings, Volume 4401
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
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- Publication Date: 2005-04-19
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Erich Novak, Stephan Dahlke, and Winfried Sickel. Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)
https://doi.org/10.4230/DagSemProc.04401.12
https://doi.org/10.4230/DagSemProc.04401.12
Abstract
We study the optimal approximation of the solution
of an operator equation Au=f by linear mappings of
rank n and compare this with the best n-term
approximation with respect to an optimal Riesz
basis. We consider worst case errors, where f
is an element of the unit ball of a Hilbert space.
We apply our results to boundary value problems
for elliptic PDEs on an arbitrary bounded
Lipschitz domain. Here we prove that approximation
by linear mappings is as good as the best n-term
approximation with respect to an optimal Riesz
basis. Our results are concerned with
approximation, not with computation.
Our goal is to understand better the possibilities
of nonlinear approximation.
Keywords
- elliptic operator equation
- worst case error
- linear approximation method
- nonlinear approximation method
- best n-term approximation Bernstein widths
- manifold widths
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