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Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

Authors Erich Novak, Stephan Dahlke, Winfried Sickel

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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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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.

Cite As Get BibTex

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

Author Details

Erich Novak
Stephan Dahlke
Winfried Sickel
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