TY - CHAP
AU - Kuklinski, Christiane
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
ID - 8577
T2 - Beiträge zum Mathematikunterricht 2019
TI - Erklärung von Mathematikleistung im Ingenieursstudium
ER -
TY - JOUR
AB - Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.
AU - McLachlan, Robert I
AU - Offen, Christian
AU - Tapley, Benjamin K
ID - 19945
IS - 1
JF - Journal of Computational Dynamics
SN - 2158-2505
TI - Symplectic integration of PDEs using Clebsch variables
VL - 6
ER -
TY - JOUR
AU - Vieluf, Solveig
AU - Mora, Karin
AU - Gölz, Christian
AU - Reuter, Eva-Maria
AU - Godde, Ben
AU - Dellnitz, Michael
AU - Reinsberger, Claus
AU - Voelcker-Rehage, Claudia
ID - 16714
JF - Neuroscience
SN - 0306-4522
TI - Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control
ER -
TY - CHAP
AU - Liebendörfer, Michael
ED - Didaktik der Mathematik der Universität Paderborn, Fachgruppe
ID - 8573
T2 - Beiträge zum Mathematikunterricht 2018
TI - Psychologische Grundbedürfnisse im frühen Mathematikstudium
ER -
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 - JOUR
AU - Klus, Stefan
AU - Nüske, Feliks
AU - Koltai, Péter
AU - Wu, Hao
AU - Kevrekidis, Ioannis
AU - Schütte, Christof
AU - Noé, Frank
ID - 21941
JF - Journal of Nonlinear Science
SN - 0938-8974
TI - Data-Driven Model Reduction and Transfer Operator Approximation
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 - 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
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 - JOUR
AU - Steffen, Eckhard
ID - 10142
IS - 1
JF - Australasian Journal of Combinatorics
TI - Approximating Vizing’s independence number conjecture
VL - 71
ER -
TY - GEN
AU - Hochmuth, Reinhard
AU - Biehler, Rolf
AU - Schaper, Niclas
AU - Kuklinski, Christiane
AU - Lankeit, Elisa
AU - Leis, Elena
AU - Liebendörfer, Michael
AU - Schürmann, Mirko
ID - 16967
TI - Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathmatikbezogenes Lernen in der Studieneingangsphase : Schlussbericht: Teilprojekt A der Leibniz Universität Hannover, Teilprojekte B und C der Universität Paderborn: Berichtszeitraum: 01.03.2015-31.08.2018
ER -
TY - JOUR
AU - Bittracher, Andreas
AU - Koltai, Péter
AU - Klus, Stefan
AU - Banisch, Ralf
AU - Dellnitz, Michael
AU - Schütte, Christof
ID - 16715
JF - Journal of Nonlinear Science
SN - 0938-8974
TI - Transition Manifolds of Complex Metastable Systems
VL - 28
ER -
TY - JOUR
AB - A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19935
JF - Nonlinearity
SN - 0951-7715
TI - Bifurcation of solutions to Hamiltonian boundary value problems
ER -
TY - JOUR
AU - Boninsegna, Lorenzo
AU - Nüske, Feliks
AU - Clementi, Cecilia
ID - 21942
JF - The Journal of Chemical Physics
SN - 0021-9606
TI - Sparse learning of stochastic dynamical equations
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 - 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 - JOUR
AB - In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19943
JF - New Zealand Journal of Mathematics
KW - Hamiltonian boundary value problems
KW - singularities
KW - conformal symplectic geometry
KW - catastrophe theory
KW - conjugate loci
TI - Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
VL - 48
ER -
TY - JOUR
AU - Hruska, Eugen
AU - Abella, Jayvee R.
AU - Nüske, Feliks
AU - Kavraki, Lydia E.
AU - Clementi, Cecilia
ID - 21943
JF - The Journal of Chemical Physics
SN - 0021-9606
TI - Quantitative comparison of adaptive sampling methods for protein dynamics
ER -
TY - JOUR
AU - Jin, Ligang
AU - Mazzuoccolo, Giuseppe
AU - Steffen, Eckhard
ID - 10132
JF - Discussiones Mathematicae Graph Theory
SN - 1234-3099
TI - Cores, joins and the Fano-flow conjectures
VL - 38
ER -
TY - GEN
AB - Kernel transfer operators, which can be regarded as approximations of
transfer operators such as the Perron-Frobenius or Koopman operator in
reproducing kernel Hilbert spaces, are defined in terms of covariance and
cross-covariance operators and have been shown to be closely related to the
conditional mean embedding framework developed by the machine learning
community. The goal of this paper is to show how the dominant eigenfunctions of
these operators in combination with gradient-based optimization techniques can
be used to detect long-lived coherent patterns in high-dimensional time-series
data. The results will be illustrated using video data and a fluid flow
example.
AU - Klus, Stefan
AU - Peitz, Sebastian
AU - Schuster, Ingmar
ID - 16293
T2 - arXiv:1805.10118
TI - Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions
ER -
TY - CHAP
AB - In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.
AU - Beermann, Dennis
AU - Dellnitz, Michael
AU - Peitz, Sebastian
AU - Volkwein, Stefan
ID - 8754
SN - 9783319753188
T2 - Reduced-Order Modeling (ROM) for Simulation and Optimization
TI - Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition
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 -
TY - JOUR
AU - Frühbis-Krüger, Anne
AU - Liebendörfer, Michael
ID - 8571
IS - 63
JF - Computeralgebra-Rundbrief
TI - Minisymposium CAS in der Hochschullehre - ein Blick in die Praxis
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 - JOUR
AB - Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19937
JF - Numerical Algorithms
SN - 1017-1398
TI - Symplectic integration of boundary value problems
ER -
TY - GEN
AB - Predictive control of power electronic systems always requires a suitable
model of the plant. Using typical physics-based white box models, a trade-off
between model complexity (i.e. accuracy) and computational burden has to be
made. This is a challenging task with a lot of constraints, since the model
order is directly linked to the number of system states. Even though white-box
models show suitable performance in most cases, parasitic real-world effects
often cannot be modeled satisfactorily with an expedient computational load.
Hence, a Koopman operator-based model reduction technique is presented which
directly links the control action to the system's outputs in a black-box
fashion. The Koopman operator is a linear but infinite-dimensional operator
describing the dynamics of observables of nonlinear autonomous dynamical
systems which can be nicely applied to the switching principle of power
electronic devices. Following this data-driven approach, the model order and
the number of system states are decoupled which allows us to consider more
complex systems. Extensive experimental tests with an automotive-type permanent
magnet synchronous motor fed by an IGBT 2-level inverter prove the feasibility
of the proposed modeling technique in a finite-set model predictive control
application.
AU - Hanke, Sören
AU - Peitz, Sebastian
AU - Wallscheid, Oliver
AU - Klus, Stefan
AU - Böcker, Joachim
AU - Dellnitz, Michael
ID - 21634
T2 - arXiv:1804.00854
TI - Koopman Operator-Based Finite-Control-Set Model Predictive Control for Electrical Drives
ER -
TY - JOUR
AU - Gölz, Christian
AU - Voelcker-Rehage, Claudia
AU - Mora, Karin
AU - Reuter, Eva-Maria
AU - Godde, Ben
AU - Dellnitz, Michael
AU - Reinsberger, Claus
AU - Vieluf, Solveig
ID - 16713
JF - Frontiers in Physiology
SN - 1664-042X
TI - Improved Neural Control of Movements Manifests in Expertise-Related Differences in Force Output and Brain Network Dynamics
ER -
TY - JOUR
AU - Jurgelucks, Benjamin
AU - Claes, Leander
AU - Walther, Andrea
AU - Henning, Bernd
ID - 6571
JF - Optimization Methods and Software
SN - 1055-6788
TI - Optimization of triple-ring electrodes on piezoceramic transducers using algorithmic differentiation
ER -
TY - CHAP
AU - Frühbis-Krüger, Anne
AU - Kemper, Gregor
AU - Koepf, Wolfram
AU - Liebendörfer, Michael
ID - 8572
T2 - Beiträge zum Mathematikunterricht 2018
TI - CAS in der Hochschullehre - Ein Blick in die Praxis
ER -
TY - CONF
AB - In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.
AU - Gebken, Bennet
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 8750
SN - 1860-949X
T2 - Numerical and Evolutionary Optimization – NEO 2017
TI - A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems
ER -
TY - JOUR
AB - Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.
AU - Klus, Stefan
AU - Gelß, Patrick
AU - Peitz, Sebastian
AU - Schütte, Christof
ID - 8755
IS - 7
JF - Nonlinearity
SN - 0951-7715
TI - Tensor-based dynamic mode decomposition
VL - 31
ER -
TY - GEN
AB - Two smooth map germs are right-equivalent if and only if they generate two
Lagrangian submanifolds in a cotangent bundle which have the same contact with
the zero-section. In this paper we provide a reverse direction to this
classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a
symplectic manifold have the same contact with a third Lagrangian submanifold
if and only if the intersection problems correspond to stably right equivalent
map germs. We, therefore, obtain a correspondence between local Lagrangian
intersection problems and catastrophe theory while the classical version only
captures tangential intersections. The correspondence is defined independently
of any Lagrangian fibration of the ambient symplectic manifold, in contrast to
other classical results. Moreover, we provide an extension of the
correspondence to families of local Lagrangian intersection problems. This
gives rise to a framework which allows a natural transportation of the notions
of catastrophe theory such as stability, unfolding and (uni-)versality to the
geometric setting such that we obtain a classification of families of local
Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.
AU - Offen, Christian
ID - 19940
T2 - arXiv:1811.10165
TI - Local intersections of Lagrangian manifolds correspond to catastrophe theory
ER -
TY - GEN
AU - Feldmann, Nadine
AU - Jurgelucks, Benjamin
AU - Claes, Leander
AU - Henning, Bernd
ID - 6595
TI - Vollständige Charakterisierung von piezoelektrischen Scheiben mit Ringelektroden
ER -
TY - JOUR
AU - Litzinger, Florian
AU - Boninsegna, Lorenzo
AU - Wu, Hao
AU - Nüske, Feliks
AU - Patel, Raajen
AU - Baraniuk, Richard
AU - Noé, Frank
AU - Clementi, Cecilia
ID - 21940
JF - Journal of Chemical Theory and Computation
SN - 1549-9618
TI - Rapid Calculation of Molecular Kinetics Using Compressed Sensing
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Klus, Stefan
AU - Ziessler, Adrian
ID - 16581
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty
ER -
TY - CONF
AU - Jurgelucks, Benjamin
AU - Feldmann, Nadine
AU - Claes, Leander
AU - Henning, Bernd
AU - Walther, Andrea
ID - 6572
T2 - Proceedings of Meetings on Acoustics
TI - Material parameter determination of a piezoelectric disc with triple-ring-electrodes for increased sensitivity
ER -
TY - CONF
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
ED - Göller, Robin
ED - Biehler, Rolf
ED - Hochmuth, Reinhard
ED - Rück, Hans-Georg
ID - 8559
T2 - Didactics of Mathematics in Higher Education as a Scientific Discipline - Conference Proceedings
TI - Perceived Competence and Incompetence in the First Year of Mathematics Studies: Forms and Situations
ER -
TY - JOUR
AB - We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.
AU - Peitz, Sebastian
AU - Schäfer, Kai
AU - Ober-Blöbaum, Sina
AU - Eckstein, Julian
AU - Köhler, Ulrich
AU - Dellnitz, Michael
ID - 8756
IS - 1
JF - IFAC-PapersOnLine
SN - 2405-8963
TI - A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles
VL - 50
ER -
TY - JOUR
AU - Wu, Hao
AU - Nüske, Feliks
AU - Paul, Fabian
AU - Klus, Stefan
AU - Koltai, Péter
AU - Noé, Frank
ID - 21939
JF - The Journal of Chemical Physics
SN - 0021-9606
TI - Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations
ER -
TY - JOUR
AU - Peitz, Sebastian
AU - Schäfer, Kai
AU - Ober-Blöbaum, Sina
AU - Eckstein, Julian
AU - Köhler, Ulrich
AU - Dellnitz, Michael
ID - 16657
JF - IFAC-PapersOnLine
SN - 2405-8963
TI - A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles * *This research was funded by the German Federal Ministry of Education and Research (BMBF) within the Leading-Edge Cluster Intelligent Technical Systems OstWestfalenLippe (it’s OWL).
ER -
TY - CONF
AB - In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated into a subdivision algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 8752
SN - 1860-949X
T2 - NEO 2016
TI - Gradient-Based Multiobjective Optimization with Uncertainties
ER -
TY - CONF
AU - Schumacher, Jan
ED - Kortenkamp, Ulrich
ED - Kuzle, Ana
ID - 7767
T2 - Beiträge zum Mathematikunterricht 2017
TI - Sinnkonstruktion beim Erkunden von Mustern und Strukturen
ER -
TY - CHAP
AU - Liebendörfer, Michael
AU - Hochmuth, Reinhard
AU - Biehler, Rolf
AU - Schaper, Niclas
AU - Kuklinski, Christiane
AU - Khellaf, Sarah
AU - Colberg, Christoph
AU - Schürmann, Mirko
AU - Rothe, Lukas
ED - Dooley, T.
ED - Gueudet, Ghislaine
ID - 8570
KW - Ziele der Lehre
T2 - Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1 – 5, 2017)
TI - A framework for goal dimensions of mathematics learning support in universities
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Klus, Stefan
ID - 16540
JF - Dynamical Systems
SN - 1468-9367
TI - Sensing and control in symmetric networks
ER -
TY - JOUR
AU - Liebendörfer, Michael
AU - Schukajlow, Stanislaw
ID - 8564
IS - 3
JF - ZDM
SN - 1863-9690, 1863-9704
TI - Interest development during the first year at university: do mathematical beliefs predict interest in mathematics?
VL - 49
ER -
TY - THES
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.
Since – in contrast to the solution of a single objective optimization problem – the
Pareto set generally consists of an infinite number of solutions, the computational
effort can quickly become challenging. This is even more the case when many problems have to be solved, when the number of objectives is high, or when the objectives
are costly to evaluate. Consequently, this thesis is devoted to the identification and
exploitation of structure both in the Pareto set and the dynamics of the underlying
model as well as to the development of efficient algorithms for solving problems with
additional parameters, with a high number of objectives or with PDE-constraints.
These three challenges are addressed in three respective parts.
In the first part, predictor-corrector methods are extended to entire Pareto sets.
When certain smoothness assumptions are satisfied, then the set of parameter dependent Pareto sets possesses additional structure, i.e. it is a manifold. The tangent
space can be approximated numerically which yields a direction for the predictor
step. In the corrector step, the predicted set converges to the Pareto set at a new
parameter value. The resulting algorithm is applied to an example from autonomous
driving.
In the second part, the hierarchical structure of Pareto sets is investigated. When
considering a subset of the objectives, the resulting solution is a subset of the Pareto
set of the original problem. Under additional smoothness assumptions, the respective subsets are located on the boundary of the Pareto set of the full problem. This
way, the “skeleton” of a Pareto set can be computed and due to the exponential
increase in computing time with the number of objectives, the computations of
these subsets are significantly faster which is demonstrated using an example from
industrial laundries.
In the third part, PDE-constrained multiobjective optimal control problems are
addressed by reduced order modeling methods. Reduced order models exploit the
structure in the system dynamics, for example by describing the dynamics of only the
most energetic modes. The model reduction introduces an error in both the function values and their gradients, which has to be taken into account in the development of
algorithms. Both scalarization and set-oriented approaches are coupled with reduced
order modeling. Convergence results are presented and the numerical benefit is
investigated. The algorithms are applied to semi-linear heat flow problems as well
as to the Navier-Stokes equations.
AU - Peitz, Sebastian
ID - 10594
TI - Exploiting structure in multiobjective optimization and optimal control
ER -
TY - JOUR
AU - Nüske, Feliks
AU - Wu, Hao
AU - Prinz, Jan-Hendrik
AU - Wehmeyer, Christoph
AU - Clementi, Cecilia
AU - Noé, Frank
ID - 21938
JF - The Journal of Chemical Physics
SN - 0021-9606
TI - Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias
ER -
TY - CONF
AU - Schumacher, Jan
ID - 7765
T2 - Beiträge zum Mathematikunterricht 2016
TI - Erkunden mathematischer Strukturen anstatt Interpretation in Modellen – Ein innermathematischer Zugang zu negativen Zahlen
ER -
TY - CONF
AU - Göller, Robin
AU - Liebendörfer, Michael
ED - für Didaktik der Mathematik, Gesellschaft
ID - 8566
KW - \_tablet
KW - Abschreiben
T2 - Beiträge zum Mathematikunterricht 2016
TI - Eine alternative Einstiegsvorlesung in die Fachmathematik – Konzept und Auswirkungen
ER -