@article{19938, abstract = {{We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error. }}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Foundations of Computational Mathematics}}, number = {{6}}, pages = {{1363--1400}}, title = {{{Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}}}, doi = {{10.1007/s10208-020-09454-z}}, volume = {{20}}, year = {{2020}}, } @article{19939, author = {{Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, number = {{5}}, pages = {{2335--2363}}, title = {{{Detection of high codimensional bifurcations in variational PDEs}}}, doi = {{10.1088/1361-6544/ab7293}}, volume = {{33}}, year = {{2020}}, } @inbook{17411, abstract = {{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.}}, author = {{Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian}}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, isbn = {{9783030512637}}, issn = {{2198-4182}}, publisher = {{Springer}}, title = {{{Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach}}}, doi = {{10.1007/978-3-030-51264-4_9}}, year = {{2020}}, } @article{21819, abstract = {{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.}}, author = {{Klus, Stefan and Nüske, Feliks and Hamzi, Boumediene}}, issn = {{1099-4300}}, journal = {{Entropy}}, title = {{{Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator}}}, doi = {{10.3390/e22070722}}, year = {{2020}}, } @article{16964, author = {{Hochmuth, Reinhard and Liebendörfer, Michael and Biehler, Rolf and Eichler, Andreas}}, journal = {{Neues Handbuch Hochschullehre}}, pages = {{117--138}}, title = {{{Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm)}}}, volume = {{95}}, year = {{2020}}, } @article{10596, abstract = {{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.}}, author = {{Schütze, Oliver and Cuate, Oliver and Martín, Adanay and Peitz, Sebastian and Dellnitz, Michael}}, issn = {{0305-215X}}, journal = {{Engineering Optimization}}, number = {{5}}, pages = {{832--855}}, title = {{{Pareto Explorer: a global/local exploration tool for many-objective optimization problems}}}, doi = {{10.1080/0305215x.2019.1617286}}, volume = {{52}}, year = {{2020}}, } @article{16288, abstract = {{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.}}, author = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof}}, issn = {{0167-2789}}, journal = {{Physica D: Nonlinear Phenomena}}, title = {{{Data-driven approximation of the Koopman generator: Model reduction, system identification, and control}}}, doi = {{10.1016/j.physd.2020.132416}}, volume = {{406}}, year = {{2020}}, } @inbook{16289, abstract = {{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.}}, author = {{Peitz, Sebastian and Klus, Stefan}}, booktitle = {{Lecture Notes in Control and Information Sciences}}, isbn = {{9783030357122}}, issn = {{0170-8643}}, pages = {{257--282}}, publisher = {{Springer}}, title = {{{Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator}}}, doi = {{10.1007/978-3-030-35713-9_10}}, volume = {{484}}, year = {{2020}}, } @article{16290, abstract = {{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.}}, author = {{Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael}}, issn = {{0935-4964}}, journal = {{Theoretical and Computational Fluid Dynamics}}, pages = {{577–591}}, title = {{{Deep model predictive flow control with limited sensor data and online learning}}}, doi = {{10.1007/s00162-020-00520-4}}, volume = {{34}}, year = {{2020}}, } @article{16309, abstract = {{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.}}, author = {{Peitz, Sebastian and Otto, Samuel E. and Rowley, Clarence W.}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{2162--2193}}, title = {{{Data-Driven Model Predictive Control using Interpolated Koopman Generators}}}, doi = {{10.1137/20M1325678}}, volume = {{19}}, year = {{2020}}, } @inbook{13108, abstract = {{Diagrammatisches Schlie{\ss}en wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird h{\"{a}}ufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schlie{\ss}ens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schlie{\ss}ens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schlie{\ss}en bei Lernenden rekonstruieren l{\"{a}}sst. Als m{\"{o}}gliche Werkzeuge f{\"{u}}r eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schlie{\ss}en eines Sch{\"{u}}lerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschlie{\ss}end wird die tats{\"{a}}chliche Eignung der beiden Ans{\"{a}}tze zur Rekonstruktion diagrammatischen Schlie{\ss}ens diskutiert.}}, author = {{Schumacher, Jan and Rezat, Sebastian}}, booktitle = {{Zeichen und Sprache im Mathematikunterricht}}, editor = {{Kadunz, Gert}}, publisher = {{Springer}}, title = {{{Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge}}}, doi = {{10.1007/978-3-662-61194-4_5}}, year = {{2020}}, } @inproceedings{31873, author = {{Schumacher, Jan}}, publisher = {{LibreCat University}}, title = {{{Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential der Peirceschen Semiotik}}}, doi = {{10.17877/DE290R-21555}}, year = {{2020}}, } @article{34789, author = {{Amiri, Habib and Glöckner, Helge and Schmeding, Alexander}}, issn = {{0044-8753}}, journal = {{Archivum Mathematicum}}, keywords = {{22A22, 22E65, 22E67, 46T10, 47H30, 58D15, 58H05}}, number = {{5}}, pages = {{307–356}}, title = {{{Lie groupoids of mappings taking values in a Lie groupoid}}}, doi = {{10.5817/AM2020-5-307}}, volume = {{56}}, year = {{2020}}, } @article{34787, author = {{Glöckner, Helge and Masbough, Niku}}, issn = {{0146-4124}}, journal = {{Topology Proceedings}}, keywords = {{54B10, 54D45, 54D50}}, pages = {{35–38}}, title = {{{Products of regular locally compact spaces are k_R-spaces}}}, volume = {{55}}, year = {{2020}}, } @unpublished{34808, abstract = {{For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.}}, author = {{Glöckner, Helge}}, booktitle = {{arXiv:2006.00254}}, title = {{{Smoothing operators for vector-valued functions and extension operators}}}, year = {{2020}}, } @article{29399, author = {{Limebeer, D. J. N. and Ober-Blöbaum, Sina and Farshi, F. H.}}, journal = {{IEEE Transactions on Automatic Control}}, pages = {{1381--1396}}, title = {{{Variational integrators for dissipative systems}}}, volume = {{65(4)}}, year = {{2020}}, } @article{16297, abstract = {{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.}}, author = {{Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{7593--7618}}, title = {{{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}}}, doi = {{10.1002/rnc.5197}}, volume = {{30(17)}}, year = {{2020}}, } @article{29398, author = {{Hernández Castellanos, C. I. O. and Schütze, G. and Sun, J.-Q. and Ober-Blöbaum, Sina and Morales-Luna, G.}}, journal = {{Mathematics}}, title = {{{Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping}}}, volume = {{8(11):1959}}, year = {{2020}}, } @inproceedings{29422, author = {{Lishkova, Y. and Ober-Blöbaum, Sina and Cannon, M. and Leyendecker, S.}}, booktitle = {{Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe}}, title = {{{A multirate variational approach to simulation and optimal control for flexible spacecraft}}}, year = {{2020}}, } @inproceedings{29423, author = {{Faulwasser, T. and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, K. }}, booktitle = {{24th International Symposium on Mathematical Theory of Networks and Systems}}, title = {{{A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems}}}, year = {{2020}}, }