Improved Neural Control of Movements Manifests in Expertise-Related Differences in Force Output and Brain Network Dynamics - Gölz, Christian; Voelcker-Rehage, Claudia; Mora, Karin; Reuter, Eva-Maria; Godde, Ben; Dellnitz, Michael; Reinsberger, Claus; Vieluf, Solveig (Frontiers in Physiology)

Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control - Vieluf, Solveig; Mora, Karin; Gölz, Christian; Reuter, Eva-Maria; Godde, Ben; Dellnitz, Michael; Reinsberger, Claus; Voelcker-Rehage, Claudia (Neuroscience)

Transition Manifolds of Complex Metastable Systems - Bittracher, Andreas; Koltai, Péter; Klus, Stefan; Banisch, Ralf; Dellnitz, Michael; Schütte, Christof (Journal of Nonlinear Science)

Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data - Peitz, Sebastian (arXiv:1801.06419)
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.

Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions - Klus, Stefan; Peitz, Sebastian; Schuster, Ingmar (arXiv:1805.10118)
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.

Measures of Edge-Uncolorability of Cubic Graphs - Fiol, M. A.; Mazzuoccolo, Guiseppe; Steffen, Eckhard (The Electronic Journal of Combinatorics)
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.
Keywords: Cubic graph; Tait coloring; Snark; Boole coloring; Berge's conjecture; Tutte's 5-flow conjecture

Cores, joins and the Fano-flow conjectures - Jin, Ligang; Mazzuoccolo, Giuseppe; Steffen, Eckhard (Discussiones Mathematicae Graph Theory)

Approximating Vizing’s independence number conjecture - Steffen, Eckhard (Australasian Journal of Combinatorics)

Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling - Peitz, Sebastian; Ober-Blöbaum, Sina; Dellnitz, Michael (Acta Applicandae Mathematicae)
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.

Necessary optimality conditions for optimally controlled dissipative mechanical systems modelled through fractional derivatives - Jiménez, F.; Ober-Blöbaum, Sina (6th European Conference on Computational Mechanics)

A fractional variational approach for modelling dissipative mechanical systems continuous and discrete settings - Jiménez, F.; Ober-Blöbaum, Sina (6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2018)

Computing subfields of number fields and applications to Galois group computations - Elsenhans, Andreas-Stephan; Klüners, Jürgen (Journal of Symbolic Computation)
A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given in van Hoeij et al. (2013). This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. (2013). For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields. Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.
Keywords: Computational Mathematics; Algebra and Number Theory

One Class Genera of Lattice Chains Over Number Fields - Kirschmer, Markus; Nebe, Gabriele (Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory)
We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields. If L is a lattice in the chain and p the prime ideal dividing the index of the lattices in the chain, then the {p}-arithmetic group Aut(Lp) acts chamber transitively on the corresponding Bruhat-Tits building. So our classification provides a step forward to a complete classification of these chamber transitive groups which has been announced 1987 (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their groups over number fields and one additional building with a discrete chamber transitive group.

One-class genera of exceptional groups over number fields - Kirschmer, Markus (Journal de Théorie des Nombres de Bordeaux)
We show that exceptional algebraic groups over number fields do not admit one-class genera of parahoric groups, except in the case G₂ . For the group G₂, we enumerate all such one-class genera for the usual seven-dimensional representation.
Keywords: Algebra and Number Theory

Optimization of triple-ring electrodes on piezoceramic transducers using algorithmic differentiation - Jurgelucks, Benjamin; Claes, Leander; Walther, Andrea; Henning, Bernd (Optimization Methods and Software)

Local intersections of Lagrangian manifolds correspond to catastrophe theory - Offen, Christian (arXiv:1811.10165)
Two smooth map germs are right-equivalent if and only if they generate two Lagrangian submanifolds in a cotangent bundle which have the same contact with the zero-section. In this paper we provide a reverse direction to this classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a symplectic manifold have the same contact with a third Lagrangian submanifold if and only if the intersection problems correspond to stably right equivalent map germs. We, therefore, obtain a correspondence between local Lagrangian intersection problems and catastrophe theory while the classical version only captures tangential intersections. The correspondence is defined independently of any Lagrangian fibration of the ambient symplectic manifold, in contrast to other classical results. Moreover, we provide an extension of the correspondence to families of local Lagrangian intersection problems. This gives rise to a framework which allows a natural transportation of the notions of catastrophe theory such as stability, unfolding and (uni-)versality to the geometric setting such that we obtain a classification of families of local Lagrangian intersection problems. An application is the classification of Lagrangian boundary value problems for symplectic maps.

Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci - McLachlan, Robert I; Offen, Christian (New Zealand Journal of Mathematics)
In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.
Keywords: Hamiltonian boundary value problems; singularities; conformal symplectic geometry; catastrophe theory; conjugate loci

Numerical analysis of partial differential equations on and of evolving surfaces - Kovács, Balázs

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary - Karátson, János; Kovács, Balázs; Korotov, Sergey (IMA Journal of Numerical Analysis)
Abstract The maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary - Karátson, János; Kovács, Balázs; Korotov, Sergey (IMA Journal of Numerical Analysis)
Abstract The maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Linearly implicit full discretization of surface evolution - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces - Kovács, Balázs (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis

Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias - Nüske, Feliks; Wu, Hao; Prinz, Jan-Hendrik; Wehmeyer, Christoph; Clementi, Cecilia; Noé, Frank (The Journal of Chemical Physics)

Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations - Wu, Hao; Nüske, Feliks; Paul, Fabian; Klus, Stefan; Koltai, Péter; Noé, Frank (The Journal of Chemical Physics)

Gradient-Based Multiobjective Optimization with Uncertainties - Peitz, Sebastian; Dellnitz, Michael (NEO 2016)
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.

Material parameter determination of a piezoelectric disc with triple-ring-electrodes for increased sensitivity - Jurgelucks, Benjamin; Feldmann, Nadine; Claes, Leander; Henning, Bernd; Walther, Andrea (Proceedings of Meetings on Acoustics)

Sensing and control in symmetric networks - Dellnitz, Michael; Klus, Stefan (Dynamical Systems)

A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty - Dellnitz, Michael; Klus, Stefan; Ziessler, Adrian (SIAM Journal on Applied Dynamical Systems)

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). - Peitz, Sebastian; Schäfer, Kai; Ober-Blöbaum, Sina; Eckstein, Julian; Köhler, Ulrich; Dellnitz, Michael (IFAC-PapersOnLine)

Exploiting structure in multiobjective optimization and optimal control - Peitz, Sebastian
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.

A multiobjective MPC approach for autonomously driven electric vehicles - Peitz, Sebastian; Schäfer, Kai; Ober-Blöbaum, Sina; Eckstein, Julian; Köhler, Ulrich; Dellnitz, Michael (Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC))
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.

Variational multirate integration in discrete mechanics and optimal control - Gail, T.; Ober-Blöbaum, Sina; Leyendecker, S. (ECCOMAS Thematic Conference on Multibody Dynamics)

The constructive membership problem for discrete two-generator subgroups of SL(2,R) - Kirschmer, Markus; Rüther, Marion G. (Journal of Algebra)
We describe a practical algorithm to solve the constructive membership problem for discrete two-generator subgroups of SL₂(R) or PSL₂(R). This algorithm has been implemented in Magma for groups defined over real algebraic number fields.
Keywords: Algebra and Number Theory

Convergence of finite elements on an evolving surface driven by diffusion on the surface - Kovács, Balázs; Li, Buyang; Lubich, Christian; Power Guerra, Christian A. (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Maximum norm stability and error estimates for the evolving surface finite element method - Kovács, Balázs; Power Guerra, Christian Andreas (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis

High-order evolving surface finite element method for parabolic problems on evolving surfaces - Kovács, Balázs (IMA Journal of Numerical Analysis)
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Maximum norm stability and error estimates for the evolving surface finite element method - Kovács, Balázs; Power Guerra, Christian Andreas (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis

Modeling and simulation of metallic, particle-damped spheres for lightweight materials - Steinle, Tobias
Lightweight materials play an ever growing role in today's world. Saving on the mass of a machine will usually translate into a lower energy consumption. However, lightweight applications are prone to develop performance problems due to vibration induced by the operation of the machine. The Fraunhofer Institute for Manufacturing Technology and Advanced Materials in Dresden conducts research into the damping properties of composite materials. They are experimenting with hollow, particle filled spheres embedded in the lightweight material. Such a system is the technical motivation of this thesis. Ultimately, a numerical experiment to derive the coefficient of restitution is required. The simulation developed in this thesis is based on a discrete element method to track the individual particle and sphere trajectories. Based on a potential based approach for the particle interactions deployed in molecular dynamics, the behavior of the particles can be controlled effectively. The simulated volume is using reflecting boundaries and encloses the hollow sphere. In this work, a highly flexible memory structure was used with a linked cell approach to cope with the highly flexible mass of particles. This allows for a linear complexity of the method in regard to the particle number by reducing the computational overhead of the interaction computation. Multiple numerical experiments show the great effect the particles have on the damping behavior of the system.

Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control - Dellnitz, Michael; Eckstein, Julian; Flaßkamp, Kathrin; Friedel, Patrick; Horenkamp, Christian; Köhler, Ulrich; Ober-Blöbaum, Sina; Peitz, Sebastian; Tiemeyer, Sebastian (Progress in Industrial Mathematics at ECMI)

Variational tensor approach for approximating the rare-event kinetics of macromolecular systems - Nüske, Feliks; Schneider, Reinhold; Vitalini, Francesca; Noé, Frank (The Journal of Chemical Physics)

Multiobjective Model Predictive Control of an Industrial Laundry - Peitz, Sebastian; Gräler, Manuel; Henke, Christian; Molo, Mirko Hessel-von; Dellnitz, Michael; Trächtler, Ansgar (Procedia Technology)
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.

Increasing the Sensitivity of Impedance with Respect to Material Parameters of Triple-Ring Electrode Piezoelectric Transducers using Algorithmic Differentiation - Jurgelucks, Benjamin; Claes, Leander

Optimisation of triple-ring-electrodes on piezoceramic transducers using algorithmic differentiation - Jurgelucks, Benjamin; Claes, Leander (AD2016 The 7th International Conference on Algorithmic Differentiation)

Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control - Dellnitz, Michael; Eckstein, Julian; Flaßkamp, Kathrin; Friedel, Patrick; Horenkamp, Christian; Köhler, Ulrich; Ober-Blöbaum, Sina; Peitz, Sebastian; Tiemeyer, Sebastian (Mathematics in Industry)

On the computation of attractors for delay differential equations - Ziessler, Adrian; Molo, Mirko Hessel-Von; Dellnitz, Michael (Journal of Computational Dynamics)

A comparison of two predictive approaches to control the longitudinal dynamics of electric vehicles - Eckstein, Julian; Peitz, Sebastian; Schäfer, Kai; Friedel, Patrick; Köhler, Ulrich; Hessel von Molo, Mirko; Ober-Blöbaum, Sina; Dellnitz, Michael (Procedia Technology, 3rd International Conference on System-Integrated Intelligence: New Challenges for Product and Production Engineering)
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.

Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials - Wenger, T.; Ober-Blöbaum, Sina; Leyendecker, S. (ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering)

Constrained Galerkin variational integrators and modified constrained symplectic Runge-Kutta methods - Wenger, T.; Ober-Blöbaum, Sina; Leyendecker, S. (International Conference of Numerical Analysis and Applied Mathematics (ICNAAM))