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Publications at the institute since 1990

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Fabrici, Igor; Jendrol', Stanislav; Harant, Jochen; Soták, Roman
A note on vertex colorings of plane graphs. - In: Discussiones mathematicae. - Warsaw : De Gruyter Open, ISSN 2083-5892, Bd. 34 (2014), 4, S. 849-855

https://doi.org/10.7151/dmgt.1771
Römer, Florian; Lavrenko, Anastasia; Del Galdo, Giovanni; Hotz, Thomas; Arikan, Orhan; Thomä, Reiner S.
Sparsity order estimation for single snapshot compressed sensing. - In: 48th Asilomar Conference on Signals, Systems and Computers, 2014 : 2 - 5 Nov. 2014, Pacific Grove, California.. - Piscataway, NJ : IEEE, ISBN 978-1-4799-8298-1, (2014), S. 1220-1224

http://dx.doi.org/10.1109/ACSSC.2014.7094653
Eichfelder, Gabriele;
Vector optimization in medical engineering. - In: Mathematics without boundaries. - New York [u.a.] : Springer, ISBN 978-1-4939-1123-3, (2014), S. 181-215

This chapter is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space Rm this can be done componentwise. That corresponds to the notion of an EdgeworthPareto optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.



Doerr, Carola; Ramakrishna, G.; ,
Computing minimum cycle bases in weighted partial 2-trees in linear time. - In: Journal of graph algorithms and applications : JGAA.. - [S.l.], ISSN 15261719, Bd. 18 (2014), 3, S. 325-346

https://doi.org/10.7155/jgaa.00325
Vielitz, Martin;
Nonreversible homoclinic snaking scenarios, 2014. - Online-Ressource (PDF-Datei: III, 139 S., 2,63 MB). Ilmenau : Techn. Univ., Diss., 2014

Homoclinic Snaking ist ein spezielles Phänomen bei der Fortsetzung homokliner Orbits in der Nähe eines heteroklinen Zykels, welcher eine Gleichgewichtslage und einen periodischen Orbit verbindet. Der Begriff "Snaking" bezieht sich dabei auf die Sinusform der Fortsetzungskurven. Typischer Weise tritt dieses Phänomen in reversiblen Hamilton-Systemen auf. Dabei entsprechen die zwei Snaking-Kurven symmetrischen Homoklinen, wohingegen asymmetrische Homoklinen auf Kurvenstücken liegen, welche die beiden Snaking-Kurven verbinden. Zusammengenommen bilden die Fortsetzungskurven die Snakes-and-ladders Struktur. In dieser Arbeit wird Homoclinic Snaking in nichtreversiblen DGLs betrachtet, deren reversible Struktur (allein oder zusammen mit der Hamilton Struktur) gestört wird. Ausgangspunkt dafür ist die Arbeit von Beck et. al. (Snakes, ladders, and isolas of localised patterns). Es wird gezeigt, dass die Störung der Reversibilität geschlossene Fortsetzungskurven (Isolas) oder zwei Fortsetzungskurven, hervorrufen kann, welche alternierend den ursprünglichen sinusförmigen Fortsetzungskurven (Criss-Cross Snaking) folgen. Darüber hinaus wird Homoclinic Snaking in gewöhnlichen DGLs betrachtet, welche von Beginn an keine ausgezeichnete Struktur besitzen. Es wird untersucht, wie das Verhalten des ursprünglichen heteroklinen Zykels das Fortsetzungsverhalten bestimmt. Dabei werden die beiden Fälle unterschieden, dass der periodische Orbit positive oder negative Floquet Multiplikatoren besitzt. Des Weiteren wird ein Fortsetzungsszenario bestehend aus Isolas beschrieben. Weiterhin werden Fenichelkoordinaten in der Nähe einer 1-parametrigen Familie von periodischen Orbits, in welcher in sich die Dimension der stabilen Mannigfaltigkeit ändert, konstruiert. Dazu wird eine Foliation einer erweiterten stabilen Mannigfaltigkeit konstruiert. Es wird gezeigt, dass wenn der schwach stabile Floquet Exponent gegen Null strebt, die Foliation auch im Grenzwert glatt ist. Darüber hinaus wird ein Shilnikov Problem in der Nähe der 1-parametrigen Familie von periodischen Orbits gelöst, wenn der schwach stabile Floquet Exponent gegen Null strebt. Die Analysis basiert auf der Arbeit von Krupa et al. (Fast and slow waves in the FitzHugh-Nagumo equation).



http://www.db-thueringen.de/servlets/DocumentServlet?id=25461
Vogel, Silvia;
Random approximations in multiobjective optimization. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 27 S., 304,7 KB). . - (Preprint. - M14,12)

Often decision makers have to cope with a programming problem with unknown quantitities. Then they will estimate these quantities and solve the problem as it then appears - the 'approximate problem'. Thus there is a need to establish conditions which will ensure that the solutions to the approximate problem will come close to the solutions to the true problem in a suitable manner. Confidence sets, i.e. sets that cover the true sets with a given prescribed probability, provide useful quantitative information. In this paper we consider multiobjective problems and derive confidence sets for the sets of efficient points, weakly efficient points, and the corresponding solution sets. Besides the crucial convergence conditions for the objective and/or constraint functions, one approach for the derivation of confidence sets requires some knowledge about the true problem, which may be not available. Therefore also another method, called relaxation, is suggested. This approach works without any knowledge about the true problem. The results are applied to the Markowitz model of portfolio optimization.



http://www.db-thueringen.de/servlets/DocumentServlet?id=25361
Azizov, Tomas Ya.; Trunk, Carsten
On a class of Sturm-Liouville operators which are connected to PT symmetric problems. - In: Proceedings in applied mathematics and mechanics : PAMM.. - Weinheim [u.a.] : Wiley-VCH, ISSN 1617-7061, Bd. 14 (2014), 1, S. 991-992

http://dx.doi.org/10.1002/pamm.201410476
Worthmann, Karl; Reble, Marcus; Grüne, Lars; Allgöwer, Frank
Nonlinear MPC: the impact of sampling on closed loop stability. - In: Proceedings in applied mathematics and mechanics : PAMM.. - Weinheim [u.a.] : Wiley-VCH, ISSN 1617-7061, Bd. 14 (2014), 1, S. 911-912

http://dx.doi.org/10.1002/pamm.201410436
Eichfelder, Gabriele; Pilecka, Maria
Set approach for set optimization with variable ordering structures. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 37 S., 434,7 KB). . - (Preprint. - M14,11)

This paper aims at combining variable ordering structures with set relations in set optimization, which have been dened using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new variable set relations generalizing the relations from [16, 25] and discuss their usefulness. After analyzing the properties of the introduced relations, we dene new solution notions for set-valued optimization problems equipped with variable ordering structures and compare them with other concepts from the literature. In order to characterize the introduced solutions a nonlinear scalarization approach is used.



http://www.db-thueringen.de/servlets/DocumentServlet?id=25344
Brechtken, Stefan;
A discretization of Boltzmann's collision operator with provable convergence. - In: AIP conference proceedings. - Melville, NY : Inst, ISSN 15517616, Bd. 1628 (2014), S. 1024-1031

http://dx.doi.org/10.1063/1.4902706