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DOI: 10.4230/DagSemProc.04401.12
URN: urn:nbn:de:0030-drops-1471
URL: https://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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04401.NovakErich1.Paper.147.pdf (0.4 MB)


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: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/opus/volltexte/2005/147},
  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}
}

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


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