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 - Biasco, Luca
AU - Di Gregorio, Laura
ID - 16499
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
TI - A Birkhoff–Lewis Type Theorem for the Nonlinear Wave Equation
ER -
TY - CONF
AB - In this contribution we compare two different approaches to the implementation of a Model Predictive Controller in an electric vehicle with respect to the quality of the solution and real-time applicability. The goal is to develop an intelligent cruise control in order to extend the vehicle range, i.e. to minimize energy consumption, by computing the optimal torque profile for a given track. On the one hand, a path-based linear model with strong simplifications regarding the vehicle dynamics is used. On the other hand, a nonlinear model is employed in which the dynamics of the mechanical and electrical subsystem are modeled.
AU - Eckstein, Julian
AU - Peitz, Sebastian
AU - Schäfer, Kai
AU - Friedel, Patrick
AU - Köhler, Ulrich
AU - Molo, Mirko Hessel-von
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 8758
SN - 2212-0173
T2 - Procedia Technology
TI - A Comparison of two Predictive Approaches to Control the Longitudinal Dynamics of Electric Vehicles
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Froyland, Gary
AU - Sertl, Stefan
ID - 16553
SN - 9789810243593
T2 - Equadiff 99
TI - A Conjecture on the Existence of Isolated Eigenvalues of the Perron-Frobenius Operator
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 - CONF
AU - Schütze, Oliver
AU - Talbi, El-ghazali
AU - Pulido, Gregorio Toscano
AU - Coello, Carlos Coello
AU - Santana-Quintero, Luis Vicente
ID - 16666
SN - 1424407087
T2 - 2007 IEEE Swarm Intelligence Symposium
TI - A Memetic PSO Algorithm for Scalar Optimization Problems
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Ober-Blöbaum, Sina
AU - Post, Marcus
AU - Schütze, Oliver
AU - Thiere, Bianca
ID - 16574
JF - Celestial Mechanics and Dynamical Astronomy
SN - 0923-2958
TI - A multi-objective approach to the design of low thrust space trajectories using optimal control
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 - 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
AU - Ober-Blöbaum, Sina
AU - Seifried, Albert
ID - 16643
SN - 9783033039629
T2 - 2013 European Control Conference (ECC)
TI - A multiobjective optimization approach for optimal control problems of mechanical systems with uncertainties
ER -
TY - JOUR
AU - Witting, Katrin
AU - Schulz, Bernd
AU - Dellnitz, Michael
AU - Böcker, Joachim
AU - Fröhleke, Norbert
ID - 16678
JF - International Journal on Software Tools for Technology Transfer
SN - 1433-2779
TI - A new approach for online multiobjective optimization of mechatronic systems
ER -
TY - CHAP
AU - Schütze, Oliver
ID - 16664
SN - 0302-9743
T2 - Lecture Notes in Computer Science
TI - A New Data Structure for the Nondominance Problem in Multi-objective Optimization
ER -
TY - JOUR
AU - Dellnitz, M
AU - Melbourne, I
ID - 16542
JF - Nonlinearity
SN - 0951-7715
TI - A note on the shadowing lemma and symmetric periodic points
ER -
TY - JOUR
AU - Day, S.
AU - Junge, O.
AU - Mischaikow, K.
ID - 16527
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems
ER -
TY - JOUR
AU - Junge, Oliver
AU - Osinga, Hinke M.
ID - 16619
JF - ESAIM: Control, Optimisation and Calculus of Variations
SN - 1292-8119
TI - A set oriented approach to global optimal control
ER -
TY - JOUR
AU - Grüne, Lars
AU - Junge, Oliver
ID - 16613
JF - Systems & Control Letters
SN - 0167-6911
TI - A set oriented approach to optimal feedback stabilization
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 - JOUR
AB - In this work we present a set-oriented path following method for the computation of relative global
attractors of parameter-dependent dynamical systems. We start with an initial approximation of the
relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method.
By using previously obtained approximations of the parameter-dependent relative global attractor
we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole
subdivision procedure. We illustrate the feasibility of the set-oriented path following method by
exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence
and of large-scale atmospheric regime changes .
AU - Gerlach, Raphael
AU - Ziessler, Adrian
AU - Eckhardt, Bruno
AU - Dellnitz, Michael
ID - 16710
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Hohmann, Andreas
ID - 17015
JF - Numerische Mathematik
SN - 0029-599X
TI - A subdivision algorithm for the computation of unstable manifolds and global attractors
VL - 75
ER -
TY - JOUR
AB - The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.
AU - Krauskopf, B.
AU - Osinga, H. M.
AU - Doedel, E. J.
AU - Henderson, M. E.
AU - Guckenheimer, J.
AU - Vladimirsky, A.
AU - Dellnitz, M.
AU - Junge, O.
ID - 16627
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - A Survey of Methods for Computing (un)stable Manifolds of Vector Fields
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 - Witting, Katrin
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 16677
JF - Journal of Global Optimization
SN - 0925-5001
TI - A variational approach to define robustness for parametric multiobjective optimization problems
ER -
TY - JOUR
AU - Dellnitz, M
AU - Heinrich, C
ID - 16532
JF - Nonlinearity
SN - 0951-7715
TI - Admissible symmetry increasing bifurcations
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 - JOUR
AB - Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16535
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Almost Invariant Sets in Chua's Circuit
ER -
TY - JOUR
AU - Guder, Rabbijah
AU - Dellnitz, Michael
AU - Kreuzer, Edwin
ID - 16614
JF - Chaos, Solitons & Fractals
SN - 0960-0779
TI - An adaptive method for the approximation of the generalized cell mapping
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16536
JF - Computing and Visualization in Science
SN - 1432-9360
TI - An adaptive subdivision technique for the approximation of attractors and invariant measures
ER -
TY - JOUR
AU - Junge, Oliver
ID - 16617
JF - Dynamical Systems
SN - 1468-9367
TI - An adaptive subdivision technique for the approximation of attractors and invariant measures: proof of convergence
ER -
TY - JOUR
AU - Klus, Stefan
AU - Sahai, Tuhin
AU - Liu, Cong
AU - Dellnitz, Michael
ID - 16624
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
TI - An efficient algorithm for the parallel solution of high-dimensional differential equations
ER -
TY - JOUR
AB - In this article, we present an efficient descent method for locally Lipschitz
continuous multiobjective optimization problems (MOPs). The method is realized
by combining a theoretical result regarding the computation of descent
directions for nonsmooth MOPs with a practical method to approximate the
subdifferentials of the objective functions. We show convergence to points
which satisfy a necessary condition for Pareto optimality. Using a set of test
problems, we compare our method to the multiobjective proximal bundle method by
M\"akel\"a. The results indicate that our method is competitive while being
easier to implement. While the number of objective function evaluations is
larger, the overall number of subgradient evaluations is lower. Finally, we
show that our method can be combined with a subdivision algorithm to compute
entire Pareto sets of nonsmooth MOPs.
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 16867
JF - Journal of Optimization Theory and Applications
TI - An efficient descent method for locally Lipschitz multiobjective optimization problems
ER -
TY - CONF
AU - Timmermann, Robert
AU - Dellnitz, Michael
ID - 17048
T2 - Performance Analysis of Sport IX, Part 8, Routledge
TI - Analysis of team and player performance using recorded trajectory data
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
AU - Preis, Robert
AU - Monien, Burkhard
AU - Schamberger, Stefan
ID - 16658
SN - 2154-4573
T2 - Handbook of Approximation Algorithms and Metaheuristics
TI - Approximation Algorithms for Multilevel Graph Partitioning
ER -
TY - JOUR
AU - Demoures, Francois
AU - Gay-Balmaz, Francois
AU - Leitz, Thomas
AU - Leyendecker, Sigrid
AU - Ober-Blöbaum, Sina
AU - Ratiu, Tudor S.
ID - 16583
JF - PAMM
SN - 1617-7061
TI - Asynchronous variational Lie group integration for geometrically exact beam dynamics
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17014
JF - Schlaglichter der Forschung: Zum 75. Jahrestag der Universität Hamburg
TI - Collisions of chaotic attractors
ER -
TY - CHAP
AU - Deuflhard, Peter
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Schütte, Christof
ID - 16584
SN - 1439-7358
T2 - Computational Molecular Dynamics: Challenges, Methods, Ideas
TI - Computation of Essential Molecular Dynamics by Subdivision Techniques
ER -
TY - JOUR
AU - Dellnitz, M.
AU - Witting, K.
ID - 16545
JF - International Journal of Computing Science and Mathematics
SN - 1752-5055
TI - Computation of robust Pareto points
ER -
TY - JOUR
AU - Aston, P. J.
AU - Dellnitz, M.
ID - 16498
JF - Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
SN - 1364-5021
TI - Computation of the dominant Lyapunov exponent via spatial integration using matrix norms
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17012
IS - 3
JF - IMA Journal of Numerical Analysis
TI - Computational bifurcation of periodic solutions in systems with symmetry
VL - 12
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Werner, Bodo
ID - 16682
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
TI - Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points
ER -
TY - JOUR
AU - Gail, Tobias
AU - Leyendecker, Sigrid
AU - Ober-Blöbaum, Sina
ID - 16608
JF - PAMM
SN - 1617-7061
TI - Computing time investigations for variational multirate integration
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Preis, Robert
ID - 16543
SN - 0302-9743
T2 - Lecture Notes in Computer Science
TI - Congestion and Almost Invariant Sets in Dynamical Systems
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Padberg, Kathrin
AU - Preis, Robert
AU - Thiere, Bianca
ID - 16575
SN - 9789048198832
T2 - Nonlinear Science and Complexity
TI - Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem
ER -
TY - JOUR
AU - Sahai, Tuhin
AU - Ziessler, Adrian
AU - Klus, Stefan
AU - Dellnitz, Michael
ID - 16709
JF - Nonlinear Dynamics
SN - 0924-090X
TI - Continuous relaxations for the traveling salesman problem
ER -
TY - JOUR
AU - Flaßkamp, Kathrin
AU - Timmermann, Julia
AU - Ober-Blöbaum, Sina
AU - Trächtler, Ansgar
ID - 16597
JF - International Journal of Control
SN - 0020-7179
TI - Control strategies on stable manifolds for energy-efficient swing-ups of double pendula
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 - JOUR
AU - Schütze, Oliver
AU - Laumanns, Marco
AU - Coello Coello, Carlos A.
AU - Dellnitz, Michael
AU - Talbi, El-Ghazali
ID - 16668
JF - Journal of Global Optimization
SN - 0925-5001
TI - Convergence of stochastic search algorithms to finite size pareto set approximations
ER -
TY - CHAP
AU - Schütze, Oliver
AU - Mostaghim, Sanaz
AU - Dellnitz, Michael
AU - Teich, Jürgen
ID - 16665
SN - 0302-9743
T2 - Lecture Notes in Computer Science
TI - Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques
ER -
TY - JOUR
AU - Dellnitz, M.
AU - Sch�tze, O.
AU - Hestermeyer, T.
ID - 16684
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
TI - Covering Pareto Sets by Multilevel Subdivision Techniques
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Field, Michael
AU - Golubitsky, Martin
AU - Ma, Jun
AU - Hohmann, Andreas
ID - 16550
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Cycling Chaos
ER -
TY - JOUR
AB - We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.
AU - Klus, Stefan
AU - Nüske, Feliks
AU - Peitz, Sebastian
AU - Niemann, Jan-Hendrik
AU - Clementi, Cecilia
AU - Schütte, Christof
ID - 16288
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
VL - 406
ER -
TY - JOUR
AB - In recent years, the success of the Koopman operator in dynamical systems
analysis has also fueled the development of Koopman operator-based control
frameworks. In order to preserve the relatively low data requirements for an
approximation via Dynamic Mode Decomposition, a quantization approach was
recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control
of nonlinear dynamical systems can be realized by means of switched systems
techniques, using only a finite set of autonomous Koopman operator-based
reduced models. These individual systems can be approximated very efficiently
from data. The main idea is to transform a control system into a set of
autonomous systems for which the optimal switching sequence has to be computed.
In this article, we extend these results to continuous control inputs using
relaxation. This way, we combine the advantages of the data efficiency of
approximating a finite set of autonomous systems with continuous controls. We
show that when using the Koopman generator, this relaxation --- realized by
linear interpolation between two operators --- does not introduce any error for
control affine systems. This allows us to control high-dimensional nonlinear
systems using bilinear, low-dimensional surrogate models. The efficiency of the
proposed approach is demonstrated using several examples with increasing
complexity, from the Duffing oscillator to the chaotic fluidic pinball.
AU - Peitz, Sebastian
AU - Otto, Samuel E.
AU - Rowley, Clarence W.
ID - 16309
IS - 3
JF - SIAM Journal on Applied Dynamical Systems
TI - Data-Driven Model Predictive Control using Interpolated Koopman Generators
VL - 19
ER -
TY - JOUR
AB - The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.
AU - Bieker, Katharina
AU - Peitz, Sebastian
AU - Brunton, Steven L.
AU - Kutz, J. Nathan
AU - Dellnitz, Michael
ID - 16290
JF - Theoretical and Computational Fluid Dynamics
SN - 0935-4964
TI - Deep model predictive flow control with limited sensor data and online learning
VL - 34
ER -
TY - CONF
AU - Li, R.
AU - Pottharst, A.
AU - Frohieke, N.
AU - Becker, J.
AU - Witting, K.
AU - Dellnitz, M.
AU - Znamenshchykov, O.
AU - Feldmann, R.
ID - 16631
SN - 0780389751
T2 - Twentieth Annual IEEE Applied Power Electronics Conference and Exposition, 2005. APEC 2005.
TI - Design and implementation of a hybrid energy supply system for railway vehicles
ER -
TY - JOUR
AU - Schütze, Oliver
AU - Vasile, Massimiliano
AU - Junge, Oliver
AU - Dellnitz, Michael
AU - Izzo, Dario
ID - 16669
JF - Engineering Optimization
SN - 0305-215X
TI - Designing optimal low-thrust gravity-assist trajectories using space pruning and a multi-objective approach
ER -
TY - JOUR
AU - Froyland, Gary
AU - Dellnitz, Michael
ID - 16600
JF - SIAM Journal on Scientific Computing
SN - 1064-8275
TI - Detecting and Locating Near-Optimal Almost-Invariant Sets and Cycles
ER -
TY - CONF
AU - Thiere, Bianca
AU - Ober-Blöbaum, Sina
AU - Pergola, Pierpaolo
ID - 16675
SN - 9781624101502
T2 - AIAA/AAS Astrodynamics Specialist Conference
TI - Detecting Initial Guesses for Trajectories in the (P)CRTBP
ER -
TY - JOUR
AU - Barany, Ernest
AU - Dellnitz, Michael
AU - Golubitsky, Martin
ID - 16518
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Detecting the symmetry of attractors
ER -
TY - JOUR
AU - Froyland, Gary
AU - Padberg, Kathrin
AU - England, Matthew H.
AU - Treguier, Anne Marie
ID - 16602
JF - Physical Review Letters
SN - 0031-9007
TI - Detection of Coherent Oceanic Structures via Transfer Operators
ER -
TY - CONF
AB - In this contribution, the range extension problem of electric vehicles is addressed. To this aim, an intelligent cruise control is developed based on the formulation of an optimal control problem. Solutions of this optimal control problem are energy efficient accelerator pedal position profiles. They can be computed numerically by a direct optimal control method using sequential quadratic programming. The approach is applied to two different driving scenarios. The results show that the energy efficiency is increased by using optimal control for both an artificial and a realistic scenario.
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 - 8761
SN - 2212-0173
T2 - Procedia Technology
TI - Development of an Intelligent Cruise Control Using Optimal Control Methods
VL - 15
ER -
TY - JOUR
AU - Elsässer, Robert
AU - Monien, Burkhard
AU - Preis, Robert
ID - 16586
JF - Theory of Computing Systems
SN - 1432-4350
TI - Diffusion Schemes for Load Balancing on Heterogeneous Networks
ER -
TY - JOUR
AU - Meyer, A.
ID - 16635
JF - IFAC Proceedings Volumes
SN - 1474-6670
TI - Discontinuity Induced Bifurcations in Timed Continuous Petri Nets
ER -
TY - CONF
AB - This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.
AU - Leyendecker, Sigrid
AU - Ober-Blöbaum, Sina
AU - Marsden, Jerrold E.
AU - Ortiz, Michael
ID - 16630
SN - 079184806X
T2 - Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C
TI - Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics
ER -
TY - JOUR
AU - Demoures, F.
AU - Gay-Balmaz, F.
AU - Leyendecker, S.
AU - Ober-Blöbaum, S.
AU - Ratiu, T. S.
AU - Weinand, Y.
ID - 16582
JF - Numerische Mathematik
SN - 0029-599X
TI - Discrete variational Lie group formulation of geometrically exact beam dynamics
ER -
TY - CONF
AU - Specht, Andreas
AU - Ober-Blobaum, Sina
AU - Wallscheid, Oliver
AU - Romaus, Christoph
AU - Bocker, Joachim
ID - 16672
SN - 9781467349741
T2 - 2013 International Electric Machines & Drives Conference
TI - Discrete-time model of an IPMSM based on variational integrators
ER -
TY - CONF
AU - Flasskamp, Kathrin
AU - Murphey, Todd
AU - Ober-Blobaum, Sina
ID - 16594
SN - 9783033039629
T2 - 2013 European Control Conference (ECC)
TI - Discretized switching time optimization problems
ER -
TY - CONF
AU - Klöpper, Benjamin
AU - Podlogar, Herbert
AU - Gausemeier, Jürgen
AU - Witting, Katrin
ID - 16623
SN - 9780769532998
T2 - 2008 19th International Conference on Database and Expert Systems Applications
TI - Domain Spanning Search for the Identification of Solution Patterns for the Conceptual Design of Self-Optimizing Systems
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Scheurle, Jürgen
ID - 16544
SN - 9789401044134
T2 - Dynamics, Bifurcation and Symmetry
TI - Eigenvalue Movement for a Class of Reversible Hamiltonian Systems with Three Degrees of Freedom
ER -
TY - CONF
AU - Flaskamp, K.
AU - Ober-Blobaum, S.
ID - 16589
SN - 9781457710964
T2 - 2012 American Control Conference (ACC)
TI - Energy efficient control for mechanical systems based on inherent dynamical structures
ER -
TY - CONF
AU - Knoke, Tobias
AU - Romaus, Christoph
AU - Bocker, Joachim
AU - Dell'Aere, Alessandro
AU - Witting, Katrin
ID - 16626
SN - 1553-572X
T2 - IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics
TI - Energy Management for an Onboard Storage System Based on Multi-Objective Optimization
ER -
TY - CONF
AU - Schneider, T.
AU - Schulz, B.
AU - Henke, C.
AU - Witting, K.
AU - Steenken, D.
AU - Bocker, J.
ID - 16663
SN - 9781424442515
T2 - 2009 IEEE International Electric Machines and Drives Conference
TI - Energy transfer via linear doubly-fed motor in different operating modes
ER -
TY - JOUR
AB - In real-world problems, uncertainties (e.g., errors in the measurement,
precision errors) often lead to poor performance of numerical algorithms when
not explicitly taken into account. This is also the case for control problems,
where optimal solutions can degrade in quality or even become infeasible. Thus,
there is the need to design methods that can handle uncertainty. In this work,
we consider nonlinear multi-objective optimal control problems with uncertainty
on the initial conditions, and in particular their incorporation into a
feedback loop via model predictive control (MPC). In multi-objective optimal
control, an optimal compromise between multiple conflicting criteria has to be
found. For such problems, not much has been reported in terms of uncertainties.
To address this problem class, we design an offline/online framework to compute
an approximation of efficient control strategies. This approach is closely
related to explicit MPC for nonlinear systems, where the potentially expensive
optimization problem is solved in an offline phase in order to enable fast
solutions in the online phase. In order to reduce the numerical cost of the
offline phase, we exploit symmetries in the control problems. Furthermore, in
order to ensure optimality of the solutions, we include an additional online
optimization step, which is considerably cheaper than the original
multi-objective optimization problem. We test our framework on a car
maneuvering problem where safety and speed are the objectives. The
multi-objective framework allows for online adaptations of the desired
objective. Alternatively, an automatic scalarizing procedure yields very
efficient feedback controls. Our results show that the method is capable of
designing driving strategies that deal better with uncertainties in the initial
conditions, which translates into potentially safer and faster driving
strategies.
AU - Hernández Castellanos, Carlos Ignacio
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ID - 16297
IS - 17
JF - International Journal of Robust and Nonlinear Control
TI - Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty
VL - 30
ER -
TY - JOUR
AB - Model predictive control is a prominent approach to construct a feedback
control loop for dynamical systems. Due to real-time constraints, the major
challenge in MPC is to solve model-based optimal control problems in a very
short amount of time. For linear-quadratic problems, Bemporad et al. have
proposed an explicit formulation where the underlying optimization problems are
solved a priori in an offline phase. In this article, we present an extension
of this concept in two significant ways. We consider nonlinear problems and -
more importantly - problems with multiple conflicting objective functions. In
the offline phase, we build a library of Pareto optimal solutions from which we
then obtain a valid compromise solution in the online phase according to a
decision maker's preference. Since the standard multi-parametric programming
approach is no longer valid in this situation, we instead use interpolation
between different entries of the library. To reduce the number of problems that
have to be solved in the offline phase, we exploit symmetries in the dynamical
system and the corresponding multiobjective optimal control problem. The
results are verified using two different examples from autonomous driving.
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ID - 16294
IS - 2
JF - International Journal of Robust and Nonlinear Control
TI - Explicit multiobjective model predictive control for nonlinear systems with symmetries
VL - 31
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Hohmann, Andreas
AU - Junge, Oliver
AU - Rumpf, Martin
ID - 16552
JF - Chaos: An Interdisciplinary Journal of Nonlinear Science
SN - 1054-1500
TI - Exploring invariant sets and invariant measures
ER -
TY - CHAP
AU - Froyland, Gary
ID - 16598
SN - 9781461266488
T2 - Nonlinear Dynamics and Statistics
TI - Extracting Dynamical Behavior via Markov Models
ER -
TY - CHAP
AB - In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.
AU - Peitz, Sebastian
AU - Klus, Stefan
ID - 16289
SN - 0170-8643
T2 - Lecture Notes in Control and Information Sciences
TI - Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator
VL - 484
ER -
TY - JOUR
AU - Dellnitz, M.
ID - 16556
JF - IMA Journal of Numerical Analysis
SN - 0272-4979
TI - Finding zeros by multilevel subdivision techniques
ER -
TY - CONF
AB - In comparison to classical control approaches in the field of electrical drives like the field-oriented control (FOC), model predictive control (MPC) approaches are able to provide a higher control performance. This refers to shorter settling times, lower overshoots, and a better decoupling of control variables in case of multi-variable controls. However, this can only be achieved if the used prediction model covers the actual behavior of the plant sufficiently well. In case of model deviations, the performance utilizing MPC remains below its potential. This results in effects like increased current ripple or steady state setpoint deviations. In order to achieve a high control performance, it is therefore necessary to adapt the model to the real plant behavior. When using an online system identification, a less accurate model is sufficient for commissioning of the drive system. In this paper, the combination of a finite-control-set MPC (FCS-MPC) with a system identification is proposed. The method does not require high-frequency signal injection, but uses the measured values already required for the FCS-MPC. An evaluation of the least squares-based identification on a laboratory test bench showed that the model accuracy and thus the control performance could be improved by an online update of the prediction models.
AU - Hanke, Soren
AU - Peitz, Sebastian
AU - Wallscheid, Oliver
AU - Böcker, Joachim
AU - Dellnitz, Michael
ID - 10597
SN - 9781538694145
T2 - 2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)
TI - Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification
ER -
TY - JOUR
AU - Dellnitz, M
AU - Melbourne, I
AU - Marsden, J E
ID - 16548
JF - Nonlinearity
SN - 0951-7715
TI - Generic bifurcation of Hamiltonian vector fields with symmetry
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Marsden, Jerrold E.
AU - Melbourne, Ian
AU - Scheurle, Jürgen
ID - 16547
SN - 9783034875387
T2 - Bifurcation and Symmetry
TI - Generic Bifurcations of Pendula
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Melbourne, Ian
ID - 16541
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
TI - Generic movement of eigenvalues for equivariant self-adjoint matrices
ER -
TY - JOUR
AU - Grüne, L.
AU - Junge, O.
ID - 16612
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
TI - Global Optimal Control of Perturbed Systems
ER -
TY - JOUR
AU - Chaudhuri, I.
AU - Sertl, S.
AU - Hajnal, Z.
AU - Dellnitz, M.
AU - Frauenheim, Th.
ID - 16500
JF - Applied Surface Science
SN - 0169-4332
TI - Global optimization of silicon nanoclusters
ER -
TY - JOUR
AU - Sertl, Stefan
AU - Dellnitz, Michael
ID - 16671
JF - Journal of Global Optimization
SN - 0925-5001
TI - Global Optimization using a Dynamical Systems Approach
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 - CHAP
AU - Dellnitz, Michael
AU - Molo, Mirko Hessel-von
AU - Metzner, Philipp
AU - Preis, Robert
AU - Schütte, Christof
ID - 16559
SN - 9783540356561
T2 - Analysis, Modeling and Simulation of Multiscale Problems
TI - Graph Algorithms for Dynamical Systems
ER -
TY - JOUR
AU - Ringkamp, Maik
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
AU - Schütze, Oliver
ID - 16659
JF - Engineering Optimization
SN - 0305-215X
TI - Handling high-dimensional problems with multi-objective continuation methods via successive approximation of the tangent space
ER -
TY - JOUR
AU - Campos, Cédric M.
AU - Ober-Blöbaum, Sina
AU - Trélat, Emmanuel
ID - 17041
IS - 9
JF - Discrete & Continuous Dynamical Systems - A
TI - High order variational integrators in the optimal control of mechanical systems
VL - 35
ER -
TY - JOUR
AU - Schütze, Oliver
AU - Coello Coello, Carlos A.
AU - Mostaghim, Sanaz
AU - Talbi, El-Ghazali
AU - Dellnitz, Michael
ID - 16667
JF - Engineering Optimization
SN - 0305-215X
TI - Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems
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 - Zanzottera, A.
AU - Mingotti, G.
AU - Castelli, R.
AU - Dellnitz, M.
ID - 16696
JF - Communications in Nonlinear Science and Numerical Simulation
SN - 1007-5704
TI - Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of Earth-to-halo transfers in the Sun–Earth–Moon scenario
ER -
TY - JOUR
AB - It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 16295
JF - Journal of Global Optimization
TI - Inverse multiobjective optimization: Inferring decision criteria from data
ER -
TY - JOUR
AB - We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.
AU - Peitz, Sebastian
AU - Klus, Stefan
ID - 10593
JF - Automatica
SN - 0005-1098
TI - Koopman operator-based model reduction for switched-system control of PDEs
VL - 106
ER -
TY - JOUR
AU - Padberg, Kathrin
AU - Hauff, Thilo
AU - Jenko, Frank
AU - Junge, Oliver
ID - 16648
JF - New Journal of Physics
SN - 1367-2630
TI - Lagrangian structures and transport in turbulent magnetized plasmas
ER -
TY - JOUR
AU - Padberg, Kathrin
AU - Thiere, Bianca
AU - Preis, Robert
AU - Dellnitz, Michael
ID - 16649
JF - Communications in Nonlinear Science and Numerical Simulation
SN - 1007-5704
TI - Local expansion concepts for detecting transport barriers in dynamical systems
ER -
TY - CHAP
AU - Froyland, Gary
ID - 16599
SN - 9789810243593
T2 - Equadiff 99
TI - Markov modelling for random dynamical systems
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Melbourne, Ian
ID - 16546
SN - 9783034875387
T2 - Bifurcation and Symmetry
TI - Mechanisms of Symmetry Creation
ER -
TY - CHAP
AU - Anacker, Harald
AU - Dellnitz, Michael
AU - Flaßkamp, Kathrin
AU - Groesbrink, Stefan
AU - Hartmann, Philip
AU - Heinzemann, Christian
AU - Horenkamp, Christian
AU - Kleinjohann, Bernd
AU - Kleinjohann, Lisa
AU - Korf, Sebastian
AU - Krüger, Martin
AU - Müller, Wolfgang
AU - Ober-Blöbaum, Sina
AU - Oberthür, Simon
AU - Porrmann, Mario
AU - Priesterjahn, Claudia
AU - Radkowski, Rafael
AU - Rasche, Christoph
AU - Rieke, Jan
AU - Ringkamp, Maik
AU - Stahl, Katharina
AU - Steenken, Dominik
AU - Stöcklein, Jörg
AU - Timmermann, Robert
AU - Trächtler, Ansgar
AU - Witting, Katrin
AU - Xie, Tao
AU - Ziegert, Steffen
ID - 16679
SN - 2195-4356
T2 - Lecture Notes in Mechanical Engineering
TI - Methods for the Design and Development
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Dignath, Florian
AU - Flaßkamp, Kathrin
AU - Molo, Mirko Hessel-von
AU - Krüger, Martin
AU - Timmermann, Robert
AU - Zheng, Qinghua
ID - 17035
SN - 1612-3956
T2 - Mathematics in Industry
TI - Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Dignath, Florian
AU - Flaßkamp, Kathrin
AU - Molo, Mirko Hessel-von
AU - Krüger, Martin
AU - Timmermann, Robert
AU - Zheng, Qinghua
ID - 16576
SN - 1612-3956
T2 - Mathematics in Industry
TI - Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway
ER -
TY - CONF
AU - Dell'Aere, Alessandro
ID - 16528
SN - 1553-572X
T2 - IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics
TI - Multi-Objective Optimization in Self-Optimizing Systems
ER -
TY - CONF
AU - Wang, Fang
AU - Dellnitz, Michael
ID - 16695
SN - 9781424428915
T2 - 2008 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications
TI - Multi-objective shape optimization for piezoceramics
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Schütze, Oliver
ID - 16681
SN - 9781461431275
T2 - Global Analysis of Nonlinear Dynamics
TI - Multilevel Subdivision Techniques for Scalar Optimization Problems
ER -
TY - CONF
AB - In a wide range of applications, it is desirable to optimally control a system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When it is not possible to compute the entire control trajectory in advance, for instance due to uncertainties or unforeseeable events, model predictive control methods can be applied to control the system during operation in real time. In this article, we present an algorithm for the solution of multiobjective model predictive control problems. In an offline scenario, it can be used to compute the entire set of optimal compromises whereas in a real time scenario, one optimal compromise is computed according to an operator's preference. The results are illustrated using the example of an industrial laundry. A logistics model of the laundry is developed and then utilized in the optimization routine. Results are presented for an offline as well as an online scenario.
AU - Peitz, Sebastian
AU - Gräler, Manuel
AU - Henke, Christian
AU - Molo, Mirko Hessel-von
AU - Dellnitz, Michael
AU - Trächtler, Ansgar
ID - 8759
SN - 2212-0173
T2 - Procedia Technology
TI - Multiobjective Model Predictive Control of an Industrial Laundry
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 -
TY - CONF
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 - 34
SN - 2212-0173
T2 - Progress in Industrial Mathematics at ECMI
TI - Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control
VL - 22
ER -
TY - JOUR
AB - In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.
AU - Peitz, Sebastian
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 8753
IS - 1
JF - Acta Applicandae Mathematicae
SN - 0167-8019
TI - Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling
VL - 161
ER -
TY - JOUR
AU - Ober-Blöbaum, Sina
AU - Padberg-Gehle, Kathrin
ID - 16642
JF - PAMM
SN - 1617-7061
TI - Multiobjective optimal control of fluid mixing
ER -
TY - CONF
AU - Blesken, Matthias
AU - Ruckert, Ulrich
AU - Steenken, Dominik
AU - Witting, Katrin
AU - Dellnitz, Michael
ID - 16524
SN - 9781424443109
T2 - 2009 NORCHIP
TI - Multiobjective optimization for transistor sizing of CMOS logic standard cells using set-oriented numerical techniques
ER -
TY - JOUR
AU - Geisler, M.Sc. Jens
AU - Witting, Dipl.-Math. Katrin
AU - Trächtler, Ansgar
AU - Dellnitz, Michael
ID - 16610
JF - IFAC Proceedings Volumes
SN - 1474-6670
TI - Multiobjective Optimization of Control Trajectories for the Guidance of a Rail-bound Vehicle
ER -
TY - JOUR
AB - In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100.
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 1774
IS - 1
JF - PAMM
SN - 1617-7061
TI - Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction
VL - 15
ER -
TY - CONF
AB - n this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100.
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 8760
SN - 1617-7061
T2 - PAMM
TI - Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction
ER -
TY - JOUR
AU - Bezrukov, S.
AU - Elsässer, R.
AU - Monien, B.
AU - Preis, R.
AU - Tillich, J.-P.
ID - 16521
JF - Theoretical Computer Science
SN - 0304-3975
TI - New spectral lower bounds on the bisection width of graphs
ER -
TY - CHAP
AU - Baier, Robert
AU - Molo, Mirko Hessel-von
ID - 16516
SN - 0302-9743
T2 - Large-Scale Scientific Computing
TI - Newton’s Method and Secant Method for Set-Valued Mappings
ER -
TY - JOUR
AB - AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.
AU - Melbourne, Ian
AU - Dellnitz, Michael
ID - 16633
JF - Mathematical Proceedings of the Cambridge Philosophical Society
SN - 0305-0041
TI - Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Post, Marcus
AU - Thiere, Bianca
ID - 16560
JF - Celestial Mechanics and Dynamical Astronomy
SN - 0923-2958
TI - On target for Venus – set oriented computation of energy efficient low thrust trajectories
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16537
JF - SIAM Journal on Numerical Analysis
SN - 0036-1429
TI - On the Approximation of Complicated Dynamical Behavior
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Froyland, Gary
AU - Horenkamp, Christian
AU - Padberg, Kathrin
ID - 16572
JF - GAMM-Mitteilungen
SN - 0936-7195
TI - On the Approximation of Transport Phenomena - a Dynamical Systems Approach
ER -
TY - JOUR
AU - Ziessler, Adrian
AU - Molo, Mirko Hessel-Von
AU - Dellnitz, Michael
ID - 16580
JF - Journal of Computational Dynamics
SN - 2158-2491
TI - On the computation of attractors for delay differential equations
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Gebken, Bennet
AU - Gerlach, Raphael
AU - Klus, Stefan
ID - 16712
JF - Dynamical Systems
SN - 1468-9367
TI - On the equivariance properties of self-adjoint matrices
ER -
TY - JOUR
AB - In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.
AU - Gebken, Bennet
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10595
IS - 4
JF - Journal of Global Optimization
SN - 0925-5001
TI - On the hierarchical structure of Pareto critical sets
VL - 73
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Froyland, Gary
AU - Sertl, Stefan
ID - 16554
JF - Nonlinearity
SN - 0951-7715
TI - On the isolated spectrum of the Perron-Frobenius operator
ER -
TY - CONF
AU - Gail, Tobias
AU - Leyendecker, Sigrid
AU - Ober-Blöbaum, Sina
ID - 17045
T2 - The 3rdJoint International Conference on Multibody System Dynamics
TI - On the role of quadrature rules and system dimensions in variational multirateintegrators
ER -
TY - GEN
AB - We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.
AU - Bieker, Katharina
AU - Gebken, Bennet
AU - Peitz, Sebastian
ID - 20731
T2 - arXiv:2012.07483
TI - On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation
ER -
TY - CONF
AU - Flasskamp, Kathrin
AU - Ober-Blobaum, Sina
AU - Schneider, Tobias
AU - Bocker, Joachim
ID - 16596
SN - 9781467357173
T2 - 52nd IEEE Conference on Decision and Control
TI - Optimal control of a switched reluctance drive by a direct method using a discrete variational principle
ER -
TY - JOUR
AU - Flaßkamp, Kathrin
AU - Timmermann, Julia
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
AU - Trächtler, Ansgar
ID - 16593
JF - PAMM
SN - 1617-7061
TI - Optimal Control on Stable Manifolds for a Double Pendulum
ER -
TY - JOUR
AB - We discuss nearest neighbor load balancing schemes on processor networks which are represented by a cartesian product of graphs and present a new optimal diffusion scheme for general graphs. In the first part of the paper, we introduce the Alternating-Direction load balancing scheme, which reduces the number of load balance iterations by a factor of 2 for cartesian products of graphs. The resulting flow is theoretically analyzed and can be very high for certain cases. Therefore, we further present the Mixed-Direction scheme which needs the same number of iterations but computes in most cases a much smaller flow. In the second part of the paper, we present a simple optimal diffusion scheme for general graphs, calculating a balancing flow which is minimal in the l2 norm. It is based on the spectra of the graph representing the network and needs only m-1 iterations to balance the load with m being the number of distinct eigenvalues. Known optimal diffusion schemes have the same performance, however the optimal scheme presented in this paper can be implemented in a very simple manner. The number of iterations of optimal diffusion schemes is independent of the load scenario and, thus, they are practical for networks which represent graphs with known spectra. Finally, our experiments exhibit that the new optimal scheme can successfully be combined with the Alternating-Direction and Mixed-Direction schemes for efficient load balancing on product graphs.
AU - Elsässer, Robert
AU - Monien, Burkhard
AU - Preis, Robert
AU - Frommer, Andreas
ID - 16587
JF - Parallel Processing Letters
SN - 0129-6264
TI - Optimal Diffusion Schemes and Load Balancing on Product Graphs
ER -
TY - CONF
AU - Romaus, C.
AU - Bocker, J.
AU - Witting, K.
AU - Seifried, A.
AU - Znamenshchykov, O.
ID - 16661
SN - 9781424428939
T2 - 2009 IEEE Energy Conversion Congress and Exposition
TI - Optimal energy management for a hybrid energy storage system combining batteries and double layer capacitors
ER -
TY - CONF
AU - Junge, O.
AU - Ober-Blobaum, S.
ID - 16618
SN - 0780395670
T2 - Proceedings of the 44th IEEE Conference on Decision and Control
TI - Optimal Reconfiguration of Formation Flying Satellites
ER -
TY - CONF
AU - Junge, O.
AU - Marsden, J.E.
AU - Ober-Blobaum, S.
ID - 16621
SN - 1424401712
T2 - Proceedings of the 45th IEEE Conference on Decision and Control
TI - Optimal Reconfiguration of Formation Flying Spacecraft ---a Decentralized Approach
ER -
TY - JOUR
AU - Flaßkamp, Kathrin
AU - Murphey, Todd
AU - Ober-Blöbaum, Sina
ID - 16595
JF - PAMM
SN - 1617-7061
TI - Optimization for discretized switched systems
ER -
TY - JOUR
AU - Krüger, Martin
AU - Witting, Katrin
AU - Trächtler, Ansgar
AU - Dellnitz, Michael
ID - 16629
JF - IFAC Proceedings Volumes
SN - 1474-6670
TI - Parametric Model-Order Reduction in Hierarchical Multiobjective Optimization of Mechatronic Systems*
ER -
TY - JOUR
AB - Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.
AU - Schütze, Oliver
AU - Cuate, Oliver
AU - Martín, Adanay
AU - Peitz, Sebastian
AU - Dellnitz, Michael
ID - 10596
IS - 5
JF - Engineering Optimization
SN - 0305-215X
TI - Pareto Explorer: a global/local exploration tool for many-objective optimization problems
VL - 52
ER -
TY - CONF
AU - Keuck, L.
AU - Frohleke, N.
AU - Bocker, J.
AU - Ziessler, A.
ID - 16622
SN - 9789075815221
T2 - 2015 17th European Conference on Power Electronics and Applications (EPE'15 ECCE-Europe)
TI - PFC-control for improved inductor utilization
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 - Ringkamp, Maik
AU - Ober-Blöbaum, Sina
AU - Leyendecker, Sigrid
ID - 16660
JF - PAMM
SN - 1617-7061
TI - Relaxing mixed integer optimal control problems using a time transformation
ER -
TY - JOUR
AU - Thiere, B.
ID - 16674
JF - Annals of the New York Academy of Sciences
SN - 0077-8923
TI - Return Time Dynamics as a Tool for Finding Almost Invariant Sets
ER -
TY - GEN
AB - Embedding techniques allow the approximations of finite dimensional
attractors and manifolds of infinite dimensional dynamical systems via
subdivision and continuation methods. These approximations give a topological
one-to-one image of the original set. In order to additionally reveal their
geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our
results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky
equation, as well as for the attractor of the Mackey-Glass delay differential
equation.
AU - Gerlach, Raphael
AU - Koltai, Péter
AU - Dellnitz, Michael
ID - 16711
T2 - arXiv:1902.08824
TI - Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems
ER -
TY - JOUR
AU - Froyland, Gary
AU - Junge, Oliver
AU - Ochs, Gunter
ID - 16601
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Rigorous computation of topological entropy with respect to a finite partition
ER -
TY - CHAP
AU - Junge, Oliver
ID - 16616
SN - 9789810243593
T2 - Equadiff 99
TI - Rigorous discretization of subdivision techniques
ER -
TY - GEN
AB - Multiobjective optimization plays an increasingly important role in modern
applications, where several objectives 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 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 as
is the case for models governed by partial differential equations (PDEs). To
decrease the numerical effort to an affordable amount, surrogate models can be
used to replace the expensive PDE evaluations. Existing multiobjective
optimization methods using model reduction are limited either to low parameter
dimensions or to few (ideally two) objectives. In this article, we present a
combination of the reduced basis model reduction method with a continuation
approach using inexact gradients. The resulting approach can handle an
arbitrary number of objectives while yielding a significant reduction in
computing time.
AU - Banholzer, Stefan
AU - Gebken, Bennet
AU - Dellnitz, Michael
AU - Peitz, Sebastian
AU - Volkwein, Stefan
ID - 16296
T2 - arXiv:1906.09075
TI - ROM-based multiobjective optimization of elliptic PDEs via numerical continuation
ER -
TY - CONF
AU - Jakobsmeyer, R.
AU - Schnittker, R.
AU - Timmermann, R.
AU - Zorn, R.
AU - Rückert, U.
AU - Baumeister, J.
ID - 17049
T2 - Performance Analysis of Sport IX, Part 8, Routledge
TI - Running performance analysis in basketball using recorded trajectory data
ER -
TY - CONF
AU - Krishnamurthy, A.
AU - Preis, R.
ID - 16628
SN - 0769523129
T2 - 19th IEEE International Parallel and Distributed Processing Symposium
TI - Satellite Formation, a Mobile Sensor Network in Space
ER -
TY - JOUR
AU - Dellnitz, M.
AU - Froyland, G.
AU - Horenkamp, C.
AU - Padberg-Gehle, K.
AU - Sen Gupta, A.
ID - 16573
JF - Nonlinear Processes in Geophysics
SN - 1607-7946
TI - Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators
ER -
TY - CONF
AU - Geisler, Jens
AU - Witting, Katrin
AU - Trächtler, Ansgar
AU - Dellnitz, Michael
ID - 17031
T2 - 7th International Heinz Nixdorf Symposium: Self-optimizing Mechatronic Systems: Designing the Future
TI - Self-Optimization of the Guidance Module of a Rail-bound Vehicle
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 - CONF
AU - Dellnitz, Michael
AU - Padberg, Kathrin
AU - Post, Marcus
AU - Thiere, Bianca
ID - 16561
SN - 0094-243X
T2 - AIP Conference Proceedings
TI - Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems
ER -
TY - CHAP
AU - Schütze, Oliver
AU - Witting, Katrin
AU - Ober-Blöbaum, Sina
AU - Dellnitz, Michael
ID - 16670
SN - 1860-949X
T2 - EVOLVE- A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation
TI - Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16538
SN - 1874-575X
T2 - Handbook of Dynamical Systems
TI - Set Oriented Numerical Methods for Dynamical Systems
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16539
SN - 1874-9305
T2 - Elsevier Astrodynamics Series
TI - Set Oriented Numerical Methods in Space Mission Design
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 - JOUR
AU - Flaßkamp, Kathrin
AU - Ober-Blöbaum, Sina
AU - Kobilarov, Marin
ID - 16592
JF - Journal of Nonlinear Science
SN - 0938-8974
TI - Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures
ER -
TY - JOUR
AU - Flaßkamp, Kathrin
AU - Ober-Blöbaum, Sina
AU - Kobilarov, Marin
ID - 16590
JF - PAMM
SN - 1617-7061
TI - Solving optimal control problems by using inherent dynamical properties
ER -
TY - JOUR
AU - Tantardini, Marco
AU - Fantino, Elena
AU - Ren, Yuan
AU - Pergola, Pierpaolo
AU - Gómez, Gerard
AU - Masdemont, Josep J.
ID - 16673
JF - Celestial Mechanics and Dynamical Astronomy
SN - 0923-2958
TI - Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem
ER -
TY - GEN
AU - Vasile, Massimiliano
AU - Schütze, Oliver
AU - Junge, Oliver
AU - Radice, Gimbardo
AU - Dellnitz, Michael
ID - 17028
TI - Spiral Trajectories in Global Optimisation of Interplanetary and Orbital Transfers
ER -
TY - JOUR
AB - Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (ur = muθ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral.
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Hohmann, Andreas
AU - Stewart, Ian
ID - 16551
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Spirals in Scalar Reaction–Diffusion Equations
ER -
TY - JOUR
AU - Hage-Packhäuser, Sebastian
AU - Dellnitz, Michael
ID - 17034
IS - 1
JF - Discrete & Continuous Dynamical Systems - B
TI - Stabilization via symmetry switching in hybrid dynamical systems
VL - 16
ER -
TY - CONF
AU - Flasskamp, Kathrin
AU - Murphey, Todd
AU - Ober-Blobaum, Sina
ID - 16591
SN - 9781467320665
T2 - 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)
TI - Switching time optimization in discretized hybrid dynamical systems
ER -
TY - JOUR
AU - Meyer, A.
AU - Dellnitz, M.
AU - Hessel-von Molo, M.
ID - 16639
JF - Nonlinear Analysis: Hybrid Systems
SN - 1751-570X
TI - Symmetries in timed continuous Petri nets
ER -
TY - JOUR
AU - Meyer, A.
AU - Dellnitz, M.
ID - 16636
JF - IFAC Proceedings Volumes
SN - 1474-6670
TI - Symmetries in Timed Continuous Petri Nets
ER -
TY - JOUR
AB - In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.
AU - Aston, Philip J.
AU - Dellnitz, Michael
ID - 16510
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Symmetry Breaking Bifurcations of Chaotic Attractors
ER -
TY - JOUR
AU - Flasskamp, Kathrin
AU - Hage-Packhäuser, Sebastian
AU - Ober-Blöbaum, Sina
ID - 17039
IS - 1
JF - Journal of Computational Dynamics
TI - Symmetry exploiting control of hybrid mechanical systems
VL - 2
ER -
TY - CHAP
AB - Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example.
AU - Flaßkamp, Kathrin
AU - Ober-Blöbaum, Sina
AU - Peitz, Sebastian
ED - Junge, Oliver
ED - Schütze, Oliver
ED - Froyland, Gary
ED - Ober-Blöbaum, Sina
ED - Padberg-Gehle, Kathrin
ID - 17411
SN - 2198-4182
T2 - Advances in Dynamics, Optimization and Computation
TI - Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Nicol, Matthew
ID - 16549
SN - 0066-5452
T2 - Trends and Perspectives in Applied Mathematics
TI - Symmetry of Attractors and the Karhunen-Loève Decomposition
ER -
TY - JOUR
AU - Mehta, Prashant G.
AU - Hessel-von Molo, Mirko
AU - Dellnitz, Michael
ID - 16632
JF - Journal of Difference Equations and Applications
SN - 1023-6198
TI - Symmetry of attractors and the Perron-Frobenius operator
ER -
TY - JOUR
AU - Meyer, A.
AU - Silva, M.
ID - 16638
JF - IFAC Proceedings Volumes
SN - 1474-6670
TI - Symmetry Reductions in Timed Continuous Petri Nets Under Infinite Server Semantics
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 - CHAP
AU - Dellnitz, Michael
AU - Froyland, Gary
AU - Junge, Oliver
ID - 16555
SN - 9783642625244
T2 - Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems
TI - The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems
ER -
TY - CHAP
AB - In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.
AU - Gerlach, Raphael
AU - Ziessler, Adrian
ED - Junge, Oliver
ED - Schütze, Oliver
ED - Ober-Blöbaum, Sina
ED - Padberg-Gehle, Kathrin
ID - 17994
SN - 2198-4182
T2 - Advances in Dynamics, Optimization and Computation
TI - The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems
VL - 304
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Rumpf, Martin
AU - Strzodka, Robert
ID - 17018
SN - 9789810243593
T2 - Equadiff 99
TI - The computation of an unstable invariant set inside a cylinder containing a knotted flow
ER -
TY - JOUR
AU - Baier, Robert
AU - Dellnitz, Michael
AU - Hessel-von Molo, Mirko
AU - Sertl, Stefan
AU - Kevrekidis, Ioannis G.
ID - 17043
IS - 1
JF - Journal of Computational Dynamics
TI - The computation of convex invariant sets via Newton's method
VL - 1
ER -
TY - CHAP
AU - Aston, P. J.
AU - Dellnitz, M.
ID - 16513
SN - 9789810243593
T2 - Equadiff 99
TI - The Computation of Lyapunov Exponents via Spatial Integration Using Vector Norms
ER -
TY - JOUR
AU - Aston, Philip J.
AU - Dellnitz, Michael
ID - 16511
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
TI - The computation of lyapunov exponents via spatial integration with application to blowout bifurcations
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Hohmann, Andreas
ID - 16533
SN - 9783034875202
T2 - Nonlinear Dynamical Systems and Chaos
TI - The Computation of Unstable Manifolds Using Subdivision and Continuation
ER -
TY - CHAP
AU - Golubitsky, Martin
AU - Marsden, Jerrold
AU - Stewart, Ian
AU - Dellnitz, Michael
ID - 16611
SN - 9780821803264
T2 - Normal Forms and Homoclinic Chaos
TI - The constrained Liapunov-Schmidt procedure and periodic orbits
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17013
JF - Lectures in Applied Mathematics
TI - The equivariant Darboux theorem
VL - 29
ER -
TY - JOUR
AB - In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to
the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems.
To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for
the computation of such objects of finite dimensional systems with the results obtained in the work
of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis
of partial differential equations and illustrate its feasibility by computing unstable manifolds of the
one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential
equation.
AU - Ziessler, Adrian
AU - Dellnitz, Michael
AU - Gerlach, Raphael
ID - 16708
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
TI - The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Dumitrescu, Roman
AU - Flaßkamp, Kathrin
AU - Gausemeier, Jürgen
AU - Hartmann, Philip
AU - Iwanek, Peter
AU - Korf, Sebastian
AU - Krüger, Martin
AU - Ober-Blöbaum, Sina
AU - Porrmann, Mario
AU - Priesterjahn, Claudia
AU - Stahl, Katharina
AU - Trächtler, Ansgar
AU - Vaßholz, Mareen
ID - 16577
SN - 2195-4356
T2 - Lecture Notes in Mechanical Engineering
TI - The Paradigm of Self-optimization
ER -
TY - JOUR
AU - Melbourne, Ian
AU - Dellnitz, Michael
AU - Golubitsky, Martin
ID - 16634
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
TI - The structure of symmetric attractors
ER -
TY - JOUR
AU - Froyland, Gary
AU - Horenkamp, Christian
AU - Rossi, Vincent
AU - Santitissadeekorn, Naratip
AU - Gupta, Alex Sen
ID - 16603
JF - Ocean Modelling
SN - 1463-5003
TI - Three-dimensional characterization and tracking of an Agulhas Ring
ER -
TY - CONF
AU - Day, Sarah
AU - Junge, Oliver
AU - Mischaikow, Konstantin
ID - 16525
SN - 9789812561695
T2 - EQUADIFF 2003
TI - Towards Automated Chaos Verification
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 - We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Koon, Wang Sang
AU - Lekien, Francois
AU - Lo, Martin W.
AU - Marsden, Jerrold E.
AU - Padberg, Kathrin
AU - Preis, Robert
AU - Ross, Shane D.
AU - Thiere, Bianca
ID - 16557
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Transport in Dynamical Astronomy and Multibody Problems
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Lo, Martin W.
AU - Marsden, Jerrold E.
AU - Padberg, Kathrin
AU - Preis, Robert
AU - Ross, Shane D.
AU - Thiere, Bianca
ID - 16558
JF - Physical Review Letters
SN - 0031-9007
TI - Transport of Mars-Crossing Asteroids from the Quasi-Hilda Region
ER -
TY - CONF
AU - Junge, O.
AU - Marsden, J.E.
AU - Mezic, I.
ID - 16620
SN - 0780386825
T2 - 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)
TI - Uncertainty in the dynamics of conservative maps
ER -
TY - CHAP
AU - Monien, Burkhard
AU - Preis, Robert
ID - 16640
SN - 0302-9743
T2 - Mathematical Foundations of Computer Science 2001
TI - Upper Bounds on the Bisection Width of 3- and 4-Regular Graphs
ER -
TY - JOUR
AU - Monien, Burkhard
AU - Preis, Robert
ID - 16641
JF - Journal of Discrete Algorithms
SN - 1570-8667
TI - Upper bounds on the Bisection Width of 3- and 4-regular Graphs
ER -
TY - CONF
AU - Flaßkamp, Kathrin
AU - Ober-Blöbaum, Sina
ID - 16588
SN - 9781450306294
T2 - Proceedings of the 14th international conference on Hybrid systems: computation and control - HSCC '11
TI - Variational formulation and optimal control of hybrid lagrangian systems
ER -
TY - CONF
AU - Ober-Blöbaum, Sina
AU - Lindhorst, Henning
ID - 17046
T2 - 21st International Symposium on Mathematical Theory of Networks and Systems
TI - Variational formulation and structure-preserving discretization ofnonlinear electric circuits
ER -
TY - JOUR
AU - Ober-Blöbaum, Sina
AU - Tao, Molei
AU - Cheng, Mulin
AU - Owhadi, Houman
AU - Marsden, Jerrold E.
ID - 17050
JF - Journal of Computational Physics
SN - 0021-9991
TI - Variational integrators for electric circuits
VL - 242
ER -
TY - CHAP
AU - Leitz, Thomas
AU - Ober-Blöbaum, Sina
AU - Leyendecker, Sigrid
ED - Terze, Zdravko
ID - 17044
SN - 1871-3033
T2 - Multibody Dynamics
TI - Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration
ER -
TY - JOUR
AU - Bürkle, David
AU - Dellnitz, Michael
AU - Junge, Oliver
AU - Rumpf, Martin
AU - Spielberg, Michael
ID - 17017
JF - Proceedings of Visualization 99
TI - Visualizing Complicated Dynamics
ER -