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URN: urn:nbn:de:0030-drops-1471
URL: http://drops.dagstuhl.de/opus/volltexte/2005/147/
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Novak, Erich ; Dahlke, Stephan ; Sickel, Winfried

Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

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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.

BibTeX - Entry

@InProceedings{novak_et_al:DSP:2005:147,
  author =	{Erich Novak and Stephan Dahlke and Winfried Sickel},
  title =	{Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  year =	{2005},
  editor =	{Thomas M{\"u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  number =	{04401},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2005/147},
  annote =	{Keywords: elliptic operator equation , worst case error , linear approximation method , nonlinear approximation method , best n-term approximation}
}

Keywords: elliptic operator equation , worst case error , linear approximation method , nonlinear approximation method , best n-term approximation
Freie Schlagwörter (deutsch): Bernstein widths , manifold widths
Seminar: 04401 - Algorithms and Complexity for Continuous Problems
Issue Date: 2005
Date of publication: 19.04.2005


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