@article{8751, abstract = {{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.}}, author = {{Peitz, Sebastian and Dellnitz, Michael}}, issn = {{2297-8747}}, journal = {{Mathematical and Computational Applications}}, number = {{2}}, title = {{{A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction}}}, doi = {{10.3390/mca23020030}}, volume = {{23}}, year = {{2018}}, } @inbook{8754, abstract = {{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.}}, author = {{Beermann, Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}}, booktitle = {{Reduced-Order Modeling (ROM) for Simulation and Optimization}}, isbn = {{9783319753188}}, pages = {{47--72}}, title = {{{Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition}}}, doi = {{10.1007/978-3-319-75319-5_3}}, year = {{2018}}, } @article{8755, abstract = {{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.}}, author = {{Klus, Stefan and Gelß, Patrick and Peitz, Sebastian and Schütte, Christof}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, number = {{7}}, pages = {{3359--3380}}, title = {{{Tensor-based dynamic mode decomposition}}}, doi = {{10.1088/1361-6544/aabc8f}}, volume = {{31}}, year = {{2018}}, } @inproceedings{8757, abstract = {{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.}}, author = {{Beermann, Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}}, booktitle = {{PAMM}}, issn = {{1617-7061}}, pages = {{51--54}}, title = {{{POD-based multiobjective optimal control of PDEs with non-smooth objectives}}}, doi = {{10.1002/pamm.201710015}}, year = {{2018}}, } @article{16713, author = {{Gölz, Christian and Voelcker-Rehage, Claudia and Mora, Karin and Reuter, Eva-Maria and Godde, Ben and Dellnitz, Michael and Reinsberger, Claus and Vieluf, Solveig}}, issn = {{1664-042X}}, journal = {{Frontiers in Physiology}}, title = {{{Improved Neural Control of Movements Manifests in Expertise-Related Differences in Force Output and Brain Network Dynamics}}}, doi = {{10.3389/fphys.2018.01540}}, year = {{2018}}, } @article{16714, author = {{Vieluf, Solveig and Mora, Karin and Gölz, Christian and Reuter, Eva-Maria and Godde, Ben and Dellnitz, Michael and Reinsberger, Claus and Voelcker-Rehage, Claudia}}, issn = {{0306-4522}}, journal = {{Neuroscience}}, pages = {{203--213}}, title = {{{Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control}}}, doi = {{10.1016/j.neuroscience.2018.07.025}}, year = {{2018}}, } @article{16715, author = {{Bittracher, Andreas and Koltai, Péter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Schütte, Christof}}, issn = {{0938-8974}}, journal = {{Journal of Nonlinear Science}}, pages = {{471--512}}, title = {{{Transition Manifolds of Complex Metastable Systems}}}, doi = {{10.1007/s00332-017-9415-0}}, volume = {{28}}, year = {{2018}}, } @unpublished{16292, abstract = {{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.}}, author = {{Peitz, Sebastian}}, booktitle = {{arXiv:1801.06419}}, title = {{{Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data}}}, year = {{2018}}, } @unpublished{16293, abstract = {{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.}}, author = {{Klus, Stefan and Peitz, Sebastian and Schuster, Ingmar}}, booktitle = {{arXiv:1805.10118}}, title = {{{Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions}}}, year = {{2018}}, } @article{8753, abstract = {{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.}}, author = {{Peitz, Sebastian and Ober-Blöbaum, Sina and Dellnitz, Michael}}, issn = {{0167-8019}}, journal = {{Acta Applicandae Mathematicae}}, number = {{1}}, pages = {{171–199}}, title = {{{Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling}}}, doi = {{10.1007/s10440-018-0209-7}}, volume = {{161}}, year = {{2018}}, } @article{21938, author = {{Nüske, Feliks and Wu, Hao and Prinz, Jan-Hendrik and Wehmeyer, Christoph and Clementi, Cecilia and Noé, Frank}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias}}}, doi = {{10.1063/1.4976518}}, year = {{2017}}, } @article{21939, author = {{Wu, Hao and Nüske, Feliks and Paul, Fabian and Klus, Stefan and Koltai, Péter and Noé, Frank}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations}}}, doi = {{10.1063/1.4979344}}, year = {{2017}}, } @inproceedings{8752, abstract = {{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.}}, author = {{Peitz, Sebastian and Dellnitz, Michael}}, booktitle = {{NEO 2016}}, isbn = {{9783319640624}}, issn = {{1860-949X}}, pages = {{159--182}}, title = {{{Gradient-Based Multiobjective Optimization with Uncertainties}}}, doi = {{10.1007/978-3-319-64063-1_7}}, year = {{2017}}, } @article{16540, author = {{Dellnitz, Michael and Klus, Stefan}}, issn = {{1468-9367}}, journal = {{Dynamical Systems}}, pages = {{61--79}}, title = {{{Sensing and control in symmetric networks}}}, doi = {{10.1080/14689367.2016.1215410}}, year = {{2017}}, } @article{16581, author = {{Dellnitz, Michael and Klus, Stefan and Ziessler, Adrian}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{120--138}}, title = {{{A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty}}}, doi = {{10.1137/16m1072735}}, year = {{2017}}, } @article{16657, author = {{Peitz, Sebastian and Schäfer, Kai and Ober-Blöbaum, Sina and Eckstein, Julian and Köhler, Ulrich and Dellnitz, Michael}}, issn = {{2405-8963}}, journal = {{IFAC-PapersOnLine}}, pages = {{8674--8679}}, title = {{{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).}}}, doi = {{10.1016/j.ifacol.2017.08.1526}}, year = {{2017}}, } @phdthesis{10594, abstract = {{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. }}, author = {{Peitz, Sebastian}}, title = {{{ Exploiting structure in multiobjective optimization and optimal control}}}, doi = {{10.17619/UNIPB/1-176}}, year = {{2017}}, } @article{8756, abstract = {{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.}}, author = {{Peitz, Sebastian and Schäfer, Kai and Ober-Blöbaum, Sina and Eckstein, Julian and Köhler, Ulrich and Dellnitz, Michael}}, issn = {{2405-8963}}, journal = {{Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC)}}, number = {{1}}, pages = {{8674--8679}}, title = {{{A multiobjective MPC approach for autonomously driven electric vehicles}}}, doi = {{10.1016/j.ifacol.2017.08.1526}}, volume = {{50}}, year = {{2017}}, } @inproceedings{34, author = {{Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian}}, booktitle = {{Progress in Industrial Mathematics at ECMI}}, issn = {{2212-0173}}, pages = {{633--641}}, publisher = {{Springer International Publishing}}, title = {{{Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control}}}, doi = {{10.1007/978-3-319-23413-7_87}}, volume = {{22}}, year = {{2016}}, } @article{21937, author = {{Nüske, Feliks and Schneider, Reinhold and Vitalini, Francesca and Noé, Frank}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Variational tensor approach for approximating the rare-event kinetics of macromolecular systems}}}, doi = {{10.1063/1.4940774}}, year = {{2016}}, }