Publikationen am Institut für Mathematik

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Derkach, Volodymyr; Strelnikov, Dmytro; Winkler, Henrik;
On a class of integral systems. - In: Complex analysis and operator theory. - Cham (ZG) : Springer International Publishing AG, ISSN 1661-8262, Bd. 15 (2021), 6, S. 1-39

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).



https://doi.org/10.1007/s11785-021-01148-w
Hoff, Daniel; Wendland, Holger;
A meshfree method for a PDE-constrained optimization problem. - In: SIAM journal on numerical analysis. - Philadelphia, Pa. : SIAM, 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.



https://doi.org/10.1137/20M1363510
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. - [London] : Nature Publishing Group UK, ISSN 2041-1723, Bd. 12 (2021), S. 1-16

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.



https://doi.org/10.1038/s41467-021-25207-0
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 : AT.. - Berlin : De Gruyter, 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.



https://doi.org/10.1515/auto-2021-0044
Ilchmann, Achim; Kirchhoff, Jonas;
Differential-algebraic systems are generically controllable and stabilizable. - In: Mathematics of control, signals, and systems : MCSS.. - London : Springer, ISSN 1435-568X, Bd. 33 (2021), 3, S. 359-377

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1-61. https://doi.org/10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.



https://doi.org/10.1007/s00498-021-00287-x
Jiang, Yuning; Sauerteig, Philipp; Houska, Boris; Worthmann, Karl;
Distributed optimization using ALADIN for MPC in smart grids. - In: IEEE transactions on control systems technology : a publication of the IEEE Control Systems Society.. - New York, NY : IEEE, ISSN 1558-0865, Bd. 29 (2021), 5, S. 2142-2152

This article presents a distributed optimization algorithm tailored to solve optimization problems arising in smart grids. In detail, we propose a variant of the augmented Lagrangian-based alternating direction inexact Newton (ALADIN) method, which comes along with global convergence guarantees for the considered class of linear-quadratic optimization problems. We establish local quadratic convergence of the proposed scheme and elaborate its advantages compared with the alternating direction method of multipliers (ADMM). In particular, we show that, at the cost of more communication, ALADIN requires fewer iterations to achieve the desired accuracy. Furthermore, it is numerically demonstrated that the number of iterations is independent of the number of subsystems. The effectiveness of the proposed scheme is illustrated by running both an ALADIN and an ADMM-based model predictive controller on a benchmark case study.



https://doi.org/10.1109/TCST.2020.3033010
Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane;
A steepest descent method for set optimization problems with set-valued mappings of finite cardinality. - In: Journal of optimization theory and applications. - Dordrecht [u.a.] : Springer Science + Business Media, ISSN 1573-2878, Bd. 190 (2021), 3, S. 711-743

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).



https://doi.org/10.1007/s10957-021-01887-y
Grundel, Sara M.; Heyder, Stefan; Hotz, Thomas; Ritschel, Tobias K. S.; Sauerteig, Philipp; Worthmann, Karl;
How to coordinate vaccination and social distancing to mitigate SARS-CoV-2 outbreaks. - In: SIAM journal on applied dynamical systems. - Philadelphia, Pa. : SIAM, ISSN 1536-0040, Bd. 20 (2021), 2, S. 1135-1157

Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this topic using optimization-based control on an age-differentiated compartmental model. For real-life decision-making, we investigate the impact of the planning horizon on the optimal vaccination/social distancing strategy. We find that in order to reduce social distancing in the long run, without overburdening the health care system, it is essential to vaccinate the people with the highest contact rates first. That is also the case if the objective is to minimize fatalities provided that the social distancing measures are sufficiently strict. However, for short-term planning it is optimal to focus on the high-risk group.



https://doi.org/10.1137/20M1387687
Kriesell, Matthias;
A note on uniquely 10-colorable graphs. - In: Journal of graph theory. - New York, NY [u.a.] : Wiley, ISSN 1097-0118, Bd. 98 (2021), 1, S. 24-26

Hadwiger conjectured that every graph of chromatic number k admits a clique minor of order k. Here we prove for k ≤ 10, that every graph of chromatic number k with a unique k-coloring (up to the color names) admits a clique minor of order k. The proof does not rely on the Four Color Theorem.



https://doi.org/10.1002/jgt.22679
Grüne, Lars; Schaller, Manuel; Schiela, Anton;
Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs. - In: Control, optimisation and calculus of variations : COCV.. - Les Ulis : EDP Sciences, ISSN 1262-3377, Bd. 27 (2021), S. 1-28

We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.



https://doi.org/10.1051/cocv/2021030