TY - JOUR
AB - Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 8751
IS - 2
JF - Mathematical and Computational Applications
SN - 2297-8747
TI - A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction
VL - 23
ER -
TY - CHAP
AU - Liebendörfer, Michael
AU - Kuklinski, Christiane
AU - Hochmuth, Reinhard
ED - Didaktik der Mathematik der Universität Paderborn, Fachgruppe
ID - 8574
T2 - Beiträge zum Mathematikunterricht 2018
TI - Auswirkungen von innovativen Vorlesungen für Lehramtsstudierende in der Studieneingangsphase
ER -
TY - CONF
AU - Schumacher, Jan
ID - 7766
T2 - Beiträge zum Mathematikunterricht 2018
TI - Semiotische Analyse von Sinnkonstruktionsprozessen bei einem innermathematischen Zugang zum Erlernen negativer Zahlen
ER -
TY - JOUR
AU - Steffen, Eckhard
ID - 10142
IS - 1
JF - Australasian Journal of Combinatorics
TI - Approximating Vizing’s independence number conjecture
VL - 71
ER -
TY - CONF
AB - A framework for set‐oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non‐smooth objective functionals. Using an academic example, we show that the extension to non‐smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.
AU - Beermann, Dennis
AU - Dellnitz, Michael
AU - Peitz, Sebastian
AU - Volkwein, Stefan
ID - 8757
SN - 1617-7061
T2 - PAMM
TI - POD-based multiobjective optimal control of PDEs with non-smooth objectives
ER -
TY - CONF
AB - The transition from high school to university mathematics has proven to be difficult for many students but especially for pre-service secondary teachers. To support these students at mastering this transition, various universities have introduced support measures of various kinds. The WiGeMath project developed a taxonomy that makes it possible to describe and compare these measures concerning their goals as well as their frame characteristics. We will exemplify the use of the taxonomy in the description of one specific innovative measure that was part of the WiGeMath evaluations. Moreover, we will present first results concerning the goal-fulfilment of this measure concerning affective characteristics of the student cohort and their predominant beliefs.
AU - Kuklinski, Christiane
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
AU - Biehler, Rolf
AU - Lankeit, Elisa
AU - Neuhaus, Silke
AU - Schaper, Niclas
AU - Schürmann, Mirko
ED - Durand-Guerrier, V.
ED - Hochmuth, R.
ED - Goodchild, S.
ED - Hogstad, N.M.
ID - 8575
KW - Beliefs.
KW - Motivational developments
KW - Novel approaches to teaching
KW - Teacher education
KW - Transition to and across university mathematics
T2 - Proceedings of the Second Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2018, 5-7 April 2018)
TI - Evaluating Innovative Measures in University Mathematics – The Case of Affective Outcomes in a Lecture focused on Problem-Solving
ER -
TY - JOUR
AB - There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.
AU - Fiol, M. A.
AU - Mazzuoccolo, Guiseppe
AU - Steffen, Eckhard
ID - 10129
IS - 4
JF - The Electronic Journal of Combinatorics
KW - Cubic graph
KW - Tait coloring
KW - Snark
KW - Boole coloring
KW - Berge's conjecture
KW - Tutte's 5-flow conjecture
TI - Measures of Edge-Uncolorability of Cubic Graphs
VL - 25
ER -
TY - GEN
AB - In a recent article, we presented a framework to control nonlinear partial
differential equations (PDEs) by means of Koopman operator based reduced models
and concepts from switched systems. The main idea was to transform a control
system into a set of autonomous systems for which the optimal switching
sequence has to be computed. These individual systems can be approximated very
efficiently by reduced order models obtained from data, and one can guarantee
equality of the full and the reduced objective function under certain
assumptions. In this article, we extend these results to continuous control
inputs using convex combinations of multiple Koopman operators corresponding to
constant controls, which results in a bilinear control system. Although
equality of the objectives can be carried over when the PDE depends linearly on
the control, we show that this approach is also valid in other scenarios using
several flow control examples of varying complexity.
AU - Peitz, Sebastian
ID - 16292
T2 - arXiv:1801.06419
TI - Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data
ER -
TY - BOOK
AU - Liebendörfer, Michael
ID - 8576
SN - 978-3-658-22506-3 978-3-658-22507-0
TI - Motivationsentwicklung im Mathematikstudium
ER -
TY - CHAP
AU - Biehler, Rolf
AU - Hochmuth, Reinhard
AU - Schaper, Niclas
AU - Kuklinski, Christiane
AU - Lankeit, Elisa
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Schürmann, Mirko
ED - Hanft, Anke
ED - Bischoff, Franziska
ED - Kretschmer, Stefanie
ID - 8569
T2 - 3. Auswertungsworkshop der Begleitforschung. Dokumentation der Projektbeiträge.
TI - Verbundprojekt WiGeMath: Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathematikbezogenes Lernen in der Studieneingangsphase
ER -