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The computation of convex invariant sets via Newton's method

Title data

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. Vol. 1 (June 2014) Issue 1 . - pp. 39-69.
ISSN 2158-2491
DOI: https://doi.org/10.3934/jcd.2014.1.39


Official URL: Volltext

Project information

Project title:
Project's official titleProject's id
Research Training Group "Scientific Computation: Application-Oriented Development of Models and Algorithms"GRK 693

Project financing: Deutsche Forschungsgemeinschaft
US Department of Energy

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Additional notes: 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
Subject classification: Mathematics Subject Classification Code: 65J15 (52A20 37M99 26E25 54C60)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields > Advanced Fields > Nonlinear Dynamics
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Profile Fields
Profile Fields > Advanced Fields
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 03 Mar 2015 15:25
Last Modified: 09 Mar 2016 07:07
URI: https://eref.uni-bayreuth.de/id/eprint/7789