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 -