A note on vertex colorings of plane graphs. - In: Discussiones mathematicae, ISSN 2083-5892, Bd. 34 (2014), 4, S. 849-855
https://doi.org/10.7151/dmgt.1771
Sparsity order estimation for single snapshot compressed sensing. - In: 48th Asilomar Conference on Signals, Systems and Computers, 2014, ISBN 978-1-4799-8298-1, (2014), S. 1220-1224
http://dx.doi.org/10.1109/ACSSC.2014.7094653
Vector optimization in medical engineering. - In: Mathematics without boundaries, (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.
Computing minimum cycle bases in weighted partial 2-trees in linear time. - In: Journal of graph algorithms and applications, ISSN 1526-1719, Bd. 18 (2014), 3, S. 325-346
https://doi.org/10.7155/jgaa.00325
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
On a class of Sturm-Liouville operators which are connected to PT symmetric problems. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 14 (2014), 1, S. 991-992
http://dx.doi.org/10.1002/pamm.201410476
Nonlinear MPC: the impact of sampling on closed loop stability. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 14 (2014), 1, S. 911-912
http://dx.doi.org/10.1002/pamm.201410436
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
A discretization of Boltzmann's collision operator with provable convergence. - In: AIP conference proceedings, ISSN 1551-7616, Bd. 1628 (2014), S. 1024-1031
http://dx.doi.org/10.1063/1.4902706
Translation invariant kinetic models on integer lattices. - In: AIP conference proceedings, ISSN 1551-7616, Bd. 1628 (2014), S. 640-647
http://dx.doi.org/10.1063/1.4902653