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The computation of convex invariant sets via Newton's method


Baier, Robert ; Dellnitz, Michael ; Hessel-von Molo, Mirko ; Kevrekidis, Yannis G. ; Sertl, Stefan:
The computation of convex invariant sets via Newton's method.
In: Journal of Computational Dynamics. Bd. 1 (Juni 2014) Heft 1 . - S. 39-69.
ISSN 2158-2491
DOI: https://doi.org/10.3934/jcd.2014.1.39



Link zum Volltext (externe URL): Volltext

Angaben zu Projekten

Offizieller ProjekttitelProjekt-ID
Research Training Group "Scientific Computation: Application-Oriented Development of Models and Algorithms"GRK 693

Projektfinanzierung: Deutsche Forschungsgemeinschaft
US Department of Energy


In this paper we present a novel approach to the computation of convex invariant sets of dynamical systems. Employing a Banach space formalism to describe differences of convex compact subsets of ℝⁿ [$\R^n$] by directed sets, we are able to formulate the property of a convex, compact set to be invariant as a zero-finding problem in this Banach space. We need either the additional restrictive assumption that the image of sets from a subclass of convex compacts under the dynamics remains convex or we have to convexify these images. In both cases we can apply the Newton's method in Banach spaces to approximate such invariant sets if an appropriate smoothness of a set-valued map holds. The theoretical foundations for realizing this approach are analyzed, and it is illustrated first by analytical and then by numerical examples.

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Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
Zusätzliche Informationen: Contents:
1. Introduction
2. Preliminaries
2.1 Banach spaces of directed sets
2.2 Newton iterations in Banach spaces
3. Differentiation of maps of directed sets
4. Computation of convex invariant sets by Newton's method
4.1 Concept
4.2 Analytical example
5 Numerical realization of the set-valued Newton's method
5.1 Approximation of directed sets
5.2 Realization of the Newton step
6. Numerical examples
7. Conclusion
Submitted in July 2012 as revised version of the technical report from May 2010 (University of Paderborn, 21 pages).
Keywords: Nvariant sets; set-valued Newton's method; Newton's method in Banach spaces; directed sets
Fachklassifikationen: Mathematics Subject Classification Code: 65J15 (52A20 37M99 26E25 54C60)
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik V (Angewandte Mathematik)
Profilfelder > Advanced Fields > Nichtlineare Dynamik
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Profilfelder > Advanced Fields
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 03 Mär 2015 15:25
Letzte Änderung: 09 Mär 2016 07:07
URI: https://eref.uni-bayreuth.de/id/eprint/7789