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Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

Authors: Erich Novak, Stephan Dahlke, and Winfried Sickel

Published in: Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005)

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.

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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)

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@InProceedings{novak_et_al:DagSemProc.04401.12,
  author =	{Novak, Erich and Dahlke, Stephan and Sickel, Winfried},
  title =	{{Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4401},
  editor =	{Thomas M\"{u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.12},
  URN =		{urn:nbn:de:0030-drops-1471},
  doi =		{10.4230/DagSemProc.04401.12},
  annote =	{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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DOI: 10.4230/DagSemProc.04401.13

Optimal Approximation of Elliptic Problems II: Wavelet Methods

Authors: Stephan Dahlke, Erich Novak, and Winfried Sickel

Published in: Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005)

Abstract

This talk is concerned with optimal approximations of the solutions of elliptic boundary value problems. After briefly recalling the fundamental concepts of optimality, we shall especially discuss best n-term approximation schemes based on wavelets. We shall mainly be concerned with the Poisson equation in Lipschitz domains. It turns out that wavelet schemes are suboptimal in general, but nevertheless they are superior to the usual uniform approximation methods. Moreover, for specific domains, i.e., for polygonal domains, wavelet methods are in fact optimal. These results are based on regularity estimates of the exact solution in a specific scale of Besov spaces.

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Stephan Dahlke, Erich Novak, and Winfried Sickel. Optimal Approximation of Elliptic Problems II: Wavelet Methods. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{dahlke_et_al:DagSemProc.04401.13,
  author =	{Dahlke, Stephan and Novak, Erich and Sickel, Winfried},
  title =	{{Optimal Approximation of Elliptic Problems II: Wavelet Methods}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4401},
  editor =	{Thomas M\"{u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.13},
  URN =		{urn:nbn:de:0030-drops-1381},
  doi =		{10.4230/DagSemProc.04401.13},
  annote =	{Keywords: Elliptic operator equations , worst case error , best n-term approximation , wavelets , Besov regularity}
}

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Documents authored by Sickel, Winfried

Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

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Optimal Approximation of Elliptic Problems II: Wavelet Methods

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