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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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Erich Novak
Stephan Dahlke
Winfried Sickel

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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

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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