Publications at the Institute of Mathematics

Results: 1891
Created on: Fri, 03 Dec 2021 23:10:13 +0100 in 0.0956 sec

Faulwasser, Timm; Flaßkamp, Kathrin; Ober-Blöbaum, Sina; Worthmann, Karl;
A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 54 (2021), 9, S. 624-629

Turnpikes have recently gained significant research interest in optimal control, since they allow for pivotal insights into the structure of solutions to optimal control problems. So far, mainly steady state solutions which serve as optimal operation points, are studied. This is in contrast to time-varying turnpikes, which are in the focus of this paper. More concretely, we analyze symmetry-induced velocity turnpikes, i.e. controlled relative equilibria, called trim primitives, which are optimal operation points regarding the given cost criterion. We characterize velocity turnpikes by means of dissipativity inequalities. Moreover, we study the equivalence between optimal control problems and steady-state problems via the corresponding necessary optimality conditions. An academic example is given for illustration.
Rußwurm, Franz; Esterhuizen, Willem; Worthmann, Karl; Streif, Stefan;
On MPC without terminal conditions for dynamic non-holonomic robots. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 54 (2021), 6, S. 133-138

We consider an input-constrained differential-drive robot with actuator dynamics. For this system, we establish asymptotic stability of the origin on arbitrary compact, convex sets using Model Predictive Control (MPC) without stabilizing terminal conditions despite the presence of state constraints and actuator dynamics. We note that the problem without those two additional ingredients was essentially solved beforehand, despite the fact that the linearization is not stabilizable. We propose an approach successfully solving the task at hand by combining the theory of barriers to characterize the viability kernel and an MPC framework based on so-called cost controllability. Moreover, we present a numerical case study to derive quantitative bounds on the required length of the prediction horizon. To this end, we investigate the boundary of the viability kernel and a neighbourhood of the origin, i.e. the most interesting areas.
Schweser, Thomas; Stiebitz, Michael;
Vertex partition of hypergraphs and maximum degenerate subhypergraphs. - In: Electronic Journal of Graph Theory and Applications, ISSN 2338-2287, Bd. 9 (2021), 1, S. 1-9
Schaller, Manuel; Philipp, Friedrich; Faulwasser, Timm; Worthmann, Karl; Maschke, Bernhard;
Control of port-Hamiltonian systems with minimal energy supply. - In: European journal of control, ISSN 1435-5671, Bd. 62 (2021), S. 33-40

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.
Eichfelder, Gabriele; Rocktäschel, Stefan;
Solving set-valued optimization problems using a multiobjective approach. - In: Optimization, ISSN 1029-4945, Bd. 0 (2021), 0, S. 1-32

Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples.
Gerlach, Tobias; Rocktäschel, Stefan;
On convexity and quasiconvexity of extremal value functions in set optimization. - In: Applied set-valued analysis and optimization, ISSN 2562-7783, Bd. 3 (2021), 3, S. 293-308

We study different classes of convex and quasiconvex set-valued maps defined by means of the l-less relation and the u-less relation. The aim of this paper is to formulate necessary and especially sufficient conditions for the convexity/quasiconvexity of extremal value functions.
Derkach, Volodymyr; Strelnikov, Dmytro; Winkler, Henrik;
On a class of integral systems. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 15 (2021), 6, 103, insges. 39 S.

We study spectral problems for two-dimensional integral system with two given non-decreasing functions R, W on an interval [0, b) which is a generalization of the Krein string. Associated to this system are the maximal linear relation Tmax and the minimal linear relation Tmin in the space L2(dW) which are connected by Tmax=T*min. It is shown that the limit point condition at b for this system is equivalent to the strong limit point condition for the linear relation Tmax. In the limit circle case the Evans-Everitt condition is proved to hold on a subspace T*N of Tmax characterized by the Neumann boundary condition at b. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced. Boundary triple for the linear relation Tmax in the limit point case (and for T*N in the limit circle case) is constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of R and W and the formula relating the principal Titchmarsh-Weyl coefficients of the direct and the dual integral systems is proved. For every integral system with the principal Titchmarsh-Weyl coefficients q a canonical system is constructed so that its Titchmarsh-Weyl coefficient Q is the unwrapping transform of q: Q(z)=zq(z2).
Hoff, Daniel; Wendland, Holger;
A meshfree method for a PDE-constrained optimization problem. - In: SIAM journal on numerical analysis, ISSN 1095-7170, Bd. 59 (2021), 4, S. 1896-1917

We describe a new approximation method for solving a PDE-constrained optimization problem numerically. Our method is based on the adjoint formulation of the optimization problem, leading to a system of weakly coupled, elliptic PDEs. These equations are then solved using kernel-based collocation. We derive an error analysis and give numerical examples.
Bracher, Johannes; Wolffram, Daniel; Deuschel, Jannik; Görgen, Konstantin; Ketterer, Jakob L.; Ullrich, Alexander; Abbott, Sam; Barbarossa, Maria Vittoria; Bertsimas, Dimitris; Bhatia, Sangeeta; Bodych, Marcin; Bosse, Nikos I.; Burgard, Jan Pablo; Castro, Lauren; Fairchild, Geoffrey; Fuhrmann, Jan; Funk, Sebastian; Gogolewski, Krzysztof; Gu, Quanquan; Heyder, Stefan; Hotz, Thomas; Kheifetz, Yuri; Kirsten, Holger; Krueger, Tyll; Krymova, Ekaterina; Li, Michael Lingzhi; Meinke, Jan H.; Michaud, Isaac J.; Niedzielewski, Karol; Oża´nski, Tomasz; Rakowski, Franciszek; Scholz, Markus; Soni, Saksham; Srivastava, Ajitesh; Zieli´nski, Jakub; Zou, Difan; Gneiting, Tilmann; Schienle, Melanie;
A pre-registered short-term forecasting study of COVID-19 in Germany and Poland during the second wave. - In: Nature Communications, ISSN 2041-1723, Bd. 12 (2021), 5173, insges. 16 S.

Disease modelling has had considerable policy impact during the ongoing COVID-19 pandemic, and it is increasingly acknowledged that combining multiple models can improve the reliability of outputs. Here we report insights from ten weeks of collaborative short-term forecasting of COVID-19 in Germany and Poland (12 October-19 December 2020). The study period covers the onset of the second wave in both countries, with tightening non-pharmaceutical interventions (NPIs) and subsequently a decay (Poland) or plateau and renewed increase (Germany) in reported cases. Thirteen independent teams provided probabilistic real-time forecasts of COVID-19 cases and deaths. These were reported for lead times of one to four weeks, with evaluation focused on one- and two-week horizons, which are less affected by changing NPIs. Heterogeneity between forecasts was considerable both in terms of point predictions and forecast spread. Ensemble forecasts showed good relative performance, in particular in terms of coverage, but did not clearly dominate single-model predictions. The study was preregistered and will be followed up in future phases of the pandemic.
Hackenberg, Annika; Worthmann, Karl; Pätz, Torben; Keiner, Dörthe; Oertel, Joachim; Flaßkamp, Kathrin;
Neurochirurgische Planung mittels automatisierter Bilderkennung und optimaler Pfadplanung :
Neurosurgery planning based on automated image recognition and optimal path design. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 69 (2021), 8, S. 708-721

Stereotactic neurosurgery requires a careful planning of cannulae paths to spare eloquent areas of the brain that, if damaged, will result in loss of essential neurological function such as sensory processing, linguistic ability, vision, or motor function. We present an approach based on modelling, simulation, and optimization to set up a computational assistant tool. Thereby, we focus on the modeling of the brain topology, where we construct ellipsoidal approximations of voxel clouds based on processed MRI data. The outcome is integrated in a path-planning problem either via constraints or by penalization terms in the objective function. The surgical planning problem with obstacle avoidance is solved for different types of stereotactic cannulae using numerical simulations. We illustrate our method with a case study using real MRI data.