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 -
TY - CHAP
AU - Dellnitz, Michael
AU - Eckstein, Julian
AU - Flaßkamp, Kathrin
AU - Friedel, Patrick
AU - Horenkamp, Christian
AU - Köhler, Ulrich
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
AU - Tiemeyer, Sebastian
ID - 16579
SN - 1612-3956
T2 - Mathematics in Industry
TI - Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control
ER -