Generalized Steiner Selections Applied to Standard Problems of Set-Valued Numerical Analysis

R. Baier: Generalized Steiner Selections Applied to Standard Problems of Set-Valued Numerical Analysis
in: Differential Equations, Chaos and Variational Problems. Conference ``View on ODEs'' in Aveiro, Portugal, June 2006 (VODE 2006), V. Staicu (ed.)
Progress in Nonlinear Differential Equations and Their Application 75, Birkhäuser, Basel, 2008, 49 - 60

ISBN/ISSN/ISMV Nummer: 978-3-7643-8481-4
DOI: 10.1007/978-3-7643-8482-1_4
MR Nummer: 2409094
Zentralblattnummer: 1149.28007
Keywords: generalized Steiner selections; set-valued quadrature methods and interpolation; linear differential inclusions; attainable sets; Lipschitz and absolutely continuous selections; set operation
Mathematics Subject Classification Code: 54C65 (93B03 93C05 28B20)
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Abstract:

Generalized Steiner points and the corresponding selections for set-valued maps share interesting commutation properties with set operations which make them suitable for the set-valued numerical problems presented here. This short overview will present first applications of these selections to standard problems in this area, namely representation of convex, compact sets in |Rn and set operations, set-valued integration and interpolation as well as the calculation of attainable sets of linear differential inclusions. Hereby, the convergence results are given uniformly for a dense countable representation of generalized Steiner points/selections. To achieve this aim, stronger conditions on the set-valued map F have to be taken into account, e.g. the Lipschitz condition on F has to be satisfied for the Demyanov distance instead of the Hausdorff distance. To establish an overview on several applications, not the strongest available results are formulated in this article.

Contents:

1. Preliminaries
2. Representation and Arithmetics of Sets
3. Regularity of Set-Valued Maps
4. Set-Valued Interpolation and Quadrature Methods
5. Linear Differential Inclusions
6. Conclusions

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