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 - Hessel von Molo, Mirko AU - Ober-Blöbaum, Sina AU - Dellnitz, Michael ID - 8758 SN - 2212-0173 T2 - Procedia Technology, 3rd International Conference on System-Integrated Intelligence: New Challenges for Product and Production Engineering TI - A comparison of two predictive approaches to control the longitudinal dynamics of electric vehicles VL - 26 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 - GEN AB - The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem. AU - Sonntag, Konstantin AU - Gebken, Bennet AU - Müller, Georg AU - Peitz, Sebastian AU - Volkwein, Stefan ID - 51334 T2 - arXiv:2402.06376 TI - A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces 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 - Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC) 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 AB - AbstractApproximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions. AU - Gebken, Bennet ID - 51208 JF - Computational Optimization and Applications KW - Applied Mathematics KW - Computational Mathematics KW - Control and Optimization SN - 0926-6003 TI - A note on the convergence of deterministic gradient sampling in nonsmooth 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 - GEN AB - We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 45498 T2 - arXiv:2306.03320 TI - A parametrisation method for high-order phase reduction in coupled oscillator networks 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 - Noé, Frank AU - Nüske, Feliks ID - 21935 JF - Multiscale Modeling & Simulation SN - 1540-3459 TI - A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems 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 VL - 188 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 - Nüske, Feliks AU - Boninsegna, Lorenzo AU - Clementi, Cecilia ID - 21944 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Coarse-graining molecular systems by spectral matching 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 - THES AB - Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind. AU - Gebken, Bennet ID - 31556 TI - Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization 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 AU - Klus, Stefan AU - Nüske, Feliks AU - Koltai, Péter AU - Wu, Hao AU - Kevrekidis, Ioannis AU - Schütte, Christof AU - Noé, Frank ID - 21941 JF - Journal of Nonlinear Science SN - 0938-8974 TI - Data-Driven Model Reduction and Transfer Operator Approximation ER - TY - 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 - JOUR AB - We present a flexible trust region descend algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. The method is derivative-free in the sense that neither need derivative information be available for the expensive objectives nor are gradients approximated using repeated function evaluations as is the case in finite-difference methods. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar pendants. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points scales linearly with the parameter space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Convergence to Pareto critical points is proven and numerical examples illustrate our findings. AU - Berkemeier, Manuel Bastian AU - Peitz, Sebastian ID - 21337 IS - 2 JF - Mathematical and Computational Applications TI - Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models VL - 26 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 - 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 - JOUR AB - Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML. AU - Dellnitz, Michael AU - Hüllermeier, Eyke AU - Lücke, Marvin AU - Ober-Blöbaum, Sina AU - Offen, Christian AU - Peitz, Sebastian AU - Pfannschmidt, Karlson ID - 21600 IS - 2 JF - SIAM Journal on Scientific Computing TI - Efficient time stepping for numerical integration using reinforcement learning VL - 45 ER - TY - CHAP AB - With the ever increasing capabilities of sensors and controllers, autonomous driving is quickly becoming a reality. This disruptive change in the automotive industry poses major challenges for manufacturers as well as suppliers as entirely new design and testing strategies have to be developed to remain competitive. Most importantly, the complexity of autonomously driving vehicles in a complex, uncertain, and safety-critical environment requires new testing procedures to cover the almost infinite range of potential scenarios. AU - Peitz, Sebastian AU - Dellnitz, Michael AU - Bannenberg, Sebastian ED - Bock, H. G. ED - Küfer, K.-H. ED - Maas, P. ED - Milde, A. ED - Schulz, V. ID - 30294 SN - 1612-3956 T2 - German Success Stories in Industrial Mathematics TI - Efficient Virtual Design and Testing of Autonomous Vehicles VL - 35 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 - JOUR AU - Goelz, Christian AU - Mora, Karin AU - Stroehlein, Julia Kristin AU - Haase, Franziska Katharina AU - Dellnitz, Michael AU - Reinsberger, Claus AU - Vieluf, Solveig ID - 21195 JF - Cognitive Neurodynamics TI - Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults 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 JF - International Journal of Robust and Nonlinear Control TI - Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty VL - 30(17) 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 JF - International Journal of Robust and Nonlinear Control TI - Explicit multiobjective model predictive control for nonlinear systems with symmetries VL - 31(2) 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 - GEN AB - We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas laid out in [H. Attouch and G. Garrigos, Multiobjective optimization: an inertial approach to Pareto optima, preprint, arXiv:1506.02823, 201]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order O(t−2) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization. AU - Sonntag, Konstantin AU - Peitz, Sebastian ID - 32447 T2 - arXiv:2307.00975 TI - Fast Convergence of Inertial Multiobjective Gradient-like Systems with Asymptotic Vanishing Damping ER - TY - JOUR AB - We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, which trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first-order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent. AU - Sonntag, Konstantin AU - Peitz, Sebastian ID - 46019 JF - Journal of Optimization Theory and Applications TI - Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems 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 AB - The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. AU - Nüske, Feliks AU - Peitz, Sebastian AU - Philipp, Friedrich AU - Schaller, Manuel AU - Worthmann, Karl ID - 23428 JF - Journal of Nonlinear Science TI - Finite-data error bounds for Koopman-based prediction and control VL - 33 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 AB - Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures. AU - Bick, Christian AU - von der Gracht, Sören ID - 52726 IS - 2 JF - Journal of Complex Networks KW - Applied Mathematics KW - Computational Mathematics KW - Control and Optimization KW - Management Science and Operations Research KW - Computer Networks and Communications SN - 2051-1329 TI - Heteroclinic dynamics in network dynamical systems with higher-order interactions VL - 12 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 - GEN AB - To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady state branch "reluctant" if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49371 T2 - arXiv:2311.17186 TI - Higher order interactions lead to "reluctant" synchrony breaking 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 AB - Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49326 IS - 6 JF - SIAM Journal on Applied Mathematics KW - Applied Mathematics SN - 0036-1399 TI - Hypernetworks: Cluster Synchronization Is a Higher-Order Effect VL - 83 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 VL - 80 ER - TY - JOUR AB - Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. AU - Klus, Stefan AU - Nüske, Feliks AU - Hamzi, Boumediene ID - 21819 JF - Entropy SN - 1099-4300 TI - Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator ER - TY - JOUR AB - Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science. AU - Klus, Stefan AU - Nüske, Feliks AU - Peitz, Sebastian ID - 29673 IS - 31 JF - Journal of Physics A: Mathematical and Theoretical TI - Koopman analysis of quantum systems VL - 55 ER - TY - GEN AB - Predictive control of power electronic systems always requires a suitable model of the plant. Using typical physics-based white box models, a trade-off between model complexity (i.e. accuracy) and computational burden has to be made. This is a challenging task with a lot of constraints, since the model order is directly linked to the number of system states. Even though white-box models show suitable performance in most cases, parasitic real-world effects often cannot be modeled satisfactorily with an expedient computational load. Hence, a Koopman operator-based model reduction technique is presented which directly links the control action to the system's outputs in a black-box fashion. The Koopman operator is a linear but infinite-dimensional operator describing the dynamics of observables of nonlinear autonomous dynamical systems which can be nicely applied to the switching principle of power electronic devices. Following this data-driven approach, the model order and the number of system states are decoupled which allows us to consider more complex systems. Extensive experimental tests with an automotive-type permanent magnet synchronous motor fed by an IGBT 2-level inverter prove the feasibility of the proposed modeling technique in a finite-set model predictive control application. AU - Hanke, Sören AU - Peitz, Sebastian AU - Wallscheid, Oliver AU - Klus, Stefan AU - Böcker, Joachim AU - Dellnitz, Michael ID - 21634 T2 - arXiv:1804.00854 TI - Koopman Operator-Based Finite-Control-Set Model Predictive Control for Electrical Drives ER - TY - JOUR 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 - 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 - 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 - 16576 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 - 17035 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 - GEN AB - In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem. AU - Berkemeier, Manuel Bastian AU - Peitz, Sebastian ID - 33150 T2 - arXiv:2208.12094 TI - Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients 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 - 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 - 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 - 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 - GEN 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 - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, 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 the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization". AU - Bernreuther, Marco AU - Dellnitz, Michael AU - Gebken, Bennet AU - Müller, Georg AU - Peitz, Sebastian AU - Sonntag, Konstantin AU - Volkwein, Stefan ID - 46578 T2 - arXiv:2308.01113 TI - Multiobjective Optimization of Non-Smooth PDE-Constrained Problems 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 - GEN AB - In this article we investigate the convergence behavior of gathering protocols with fixed circulant topologies using tools form dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we model a gathering protocol as a system of ordinary differential equations whose equilibria are exactly all possible gathering points. Then, we find necessary and sufficient conditions for the structure of the underlying interaction graph such that the protocol is stable and converging, i.e., gathering, in the distributive computing sense by using tools from dynamical systems. Moreover, these tools allow for a more fine grained analysis in terms of speed of convergence in the dynamical systems sense. In fact, we derive a decomposition of the state space into stable invariant subspaces with different convergence rates. In particular, this decomposition is identical for every (linear) circulant gathering protocol, whereas only the convergence rates depend on the weights in interaction graph itself. AU - Gerlach, Raphael AU - von der Gracht, Sören AU - Dellnitz, Michael ID - 44840 KW - Dynamical Systems KW - Coupled Systems KW - Distributed Computing KW - Robot Swarms KW - Autonomous Mobile Robots KW - Gathering T2 - arXiv:2305.06632 TI - On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies ER - TY - JOUR AB - We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions. AU - Dellnitz, Michael AU - Gebken, Bennet AU - Gerlach, Raphael AU - Klus, Stefan ID - 16712 IS - 2 JF - Dynamical Systems SN - 1468-9367 TI - On the equivariance properties of self-adjoint matrices VL - 35 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 - JOUR AB - Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods. AU - Gebken, Bennet AU - Bieker, Katharina AU - Peitz, Sebastian ID - 27426 IS - 3 JF - Journal of Global Optimization TI - On the structure of regularization paths for piecewise differentiable regularization terms VL - 85 ER - TY - JOUR 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 IS - 11 JF - IEEE Transactions on Pattern Analysis and Machine Intelligence TI - On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation VL - 44 ER - TY - JOUR AB - As in almost every other branch of science, the major advances in data science and machine learning have also resulted in significant improvements regarding the modeling and simulation of nonlinear dynamical systems. It is nowadays possible to make accurate medium to long-term predictions of highly complex systems such as the weather, the dynamics within a nuclear fusion reactor, of disease models or the stock market in a very efficient manner. In many cases, predictive methods are advertised to ultimately be useful for control, as the control of high-dimensional nonlinear systems is an engineering grand challenge with huge potential in areas such as clean and efficient energy production, or the development of advanced medical devices. However, the question of how to use a predictive model for control is often left unanswered due to the associated challenges, namely a significantly higher system complexity, the requirement of much larger data sets and an increased and often problem-specific modeling effort. To solve these issues, we present a universal framework (which we call QuaSiModO: Quantization-Simulation-Modeling-Optimization) to transform arbitrary predictive models into control systems and use them for feedback control. The advantages of our approach are a linear increase in data requirements with respect to the control dimension, performance guarantees that rely exclusively on the accuracy of the predictive model, and only little prior knowledge requirements in control theory to solve complex control problems. In particular the latter point is of key importance to enable a large number of researchers and practitioners to exploit the ever increasing capabilities of predictive models for control in a straight-forward and systematic fashion. AU - Peitz, Sebastian AU - Bieker, Katharina ID - 21199 JF - Automatica TI - On the Universal Transformation of Data-Driven Models to Control Systems VL - 149 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 - Hruska, Eugen AU - Abella, Jayvee R. AU - Nüske, Feliks AU - Kavraki, Lydia E. AU - Clementi, Cecilia ID - 21943 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Quantitative comparison of adaptive sampling methods for protein dynamics ER - TY - JOUR AU - 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 - 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 - CHAP 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 ED - Michael, Hintermüller ED - Roland, Herzog ED - Christian, Kanzow ED - Michael, Ulbrich ED - Stefan, Ulbrich ID - 16296 SN - 978-3-030-79392-0 T2 - Non-Smooth and Complementarity-Based Distributed Parameter Systems 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 - JOUR AU - Boninsegna, Lorenzo AU - Nüske, Feliks AU - Clementi, Cecilia ID - 21942 JF - The Journal of Chemical Physics SN - 0021-9606 TI - Sparse learning of stochastic dynamical equations ER - TY - JOUR AB - The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting. AU - Nüske, Feliks AU - Koltai, Péter AU - Boninsegna, Lorenzo AU - Clementi, Cecilia ID - 21820 JF - Entropy SN - 1099-4300 TI - Spectral Properties of Effective Dynamics from Conditional Expectations 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 - Klus, Stefan AU - Gelß, Patrick AU - Nüske, Feliks AU - Noé, Frank ID - 24170 JF - Machine Learning: Science and Technology SN - 2632-2153 TI - Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry 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 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 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 AU - Nüske, Feliks AU - Gelß, Patrick AU - Klus, Stefan AU - Clementi, Cecilia ID - 24169 JF - Physica D: Nonlinear Phenomena SN - 0167-2789 TI - Tensor-based computation of metastable and coherent sets 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 -