Publikationen am Institut für Mathematik

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Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl
Finite-data error bounds for Koopman-based prediction and control. - In: Journal of nonlinear science, ISSN 1432-1467, Bd. 33 (2023), 1, 14, S. 1-34

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein-Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.
Behrndt, Jussi; Schmitz, Philipp; Teschl, Gerald; Trunk, Carsten
Relative oscillation theory and essential spectra of Sturm-Liouville operators. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 518 (2023), 1, 126673
Schaller, Manuel; Kleyman, Viktoria; Mordmüller, Mario; Schmidt, Christian; Wilson, Mitsuru; Brinkmann, Ralf; Müller, Matthias A.; Worthmann, Karl
Model predictive control for retinal laser treatment at 1 kHz. - In: Automatisierungstechnik, ISSN 2196-677X, Bd. 70 (2022), 11, S. 992-1002

Laser photocoagulation is a technique applied in the treatment of retinal disease, which is often done manually or using simple control schemes. We pursue an optimization-based approach, namely Model Predictive Control (MPC), to enforce bounds on the peak temperature and, thus, to ensure safety during the medical treatment procedure - despite the spot-dependent absorption of the tissue. The desired laser repetition rate of 1 kHz is renders the requirements on the computation time of the MPC feedback a major challenge. We present a tailored MPC scheme using parametric model reduction, an extended Kalman filter for the parameter and state estimation, and suitably tuned stage costs and verify its applicability both in simulation and experiments with porcine eyes. Moreover, we give some insight on the implementation specifically tailored for fast numerical computations.
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; Fiedler, Jochen; Fuhrmann, Jan; Funk, Sebastian; Gambin, Anna; Gogolewski, Krzysztof; Heyder, Stefan; Hotz, Thomas; Kheifetz, Yuri; Kirsten, Holger; Krueger, Tyll; Krymova, Ekaterina; Leithäuser, Neele; Li, Michael L.; Meinke, Jan H.; Miasojedow, Błażej; Michaud, Isaac J.; Mohring, Jan; Nouvellet, Pierre; Nowosielski, Jedrzej M.; Ozanski, Tomasz; Radwan, Maciej; Rakowski, Franciszek; Scholz, Markus; Soni, Saksham; Srivastava, Ajitesh; Gneiting, Tilmann; Schienle, Melanie
National and subnational short-term forecasting of COVID-19 in Germany and Poland during early 2021. - In: Communications medicine, ISSN 2730-664X, Bd. 2 (2022), 136, S. 1-17

During the COVID-19 pandemic there has been a strong interest in forecasts of the short-term development of epidemiological indicators to inform decision makers. In this study we evaluate probabilistic real-time predictions of confirmed cases and deaths from COVID-19 in Germany and Poland for the period from January through April 2021.
Faulwasser, Timm; Flaßkamp, Kathrin; Ober-Blöbaum, Sina; Schaller, Manuel; Worthmann, Karl
Manifold turnpikes, trims, and symmetries. - In: Mathematics of control, signals, and systems, ISSN 1435-568X, Bd. 34 (2022), 4, S. 759-788

Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced manifold coincide with those of a reduced-order problem defined on the manifold under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. Furthermore, we show how the classical Legendre transformation between Euler-Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings.
Berger, Thomas; Snoo, Hendrik S. V. de; Trunk, Carsten; Winkler, Henrik
A Jordan-like decomposition for linear relations in finite-dimensional spaces. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (34 Seiten). - (Preprint ; M22,05)

A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space H (i.e., A is a linear subspace of H × H and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product H × H), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue infinity; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.
Mühlenhoff, Julian; Körbner, Thorben; Miccoli, Giovanni; Keiner, Dörthe; Hoffmann, Matthias K.; Sauerteig, Philipp; Worthmann, Karl; Flaßkamp, Kathrin; Urbschat, Steffi; Oertel, Joachim; Sattel, Thomas
A manually actuated continuum robot research platform for deployable shape-memory curved cannulae in stereotactic neurosurgery. - In: IEEE Xplore digital library, ISSN 2473-2001, (2022), S. 10-13

In this paper, a research platform for concentric tube continuum robots is developed in order to enable advances in deploying curved cannulae for stereotactic neurosurgery. The system consists of a manually operated high-precision actuation apparatus and a photogrammetric system with measurement errors in the range of 100 micrometer. With this platform, previously planned curved paths can be analyzed ex-situ w.r.t., e.g., target precision, follow-the-leader-behavior, and hysteretic phenomena. Regarding research towards an in-vivo application in human brains, first tests with porcine brain cadavers inside an intraoperative CT are conducted in order to pave the way for histological as well as target reachability studies.
Sauerteig, Philipp; Hoffmann, Matthias K.; Mühlenhoff, Julian; Miccoli, Giovanni; Keiner, Dörthe; Urbschat, Steffi; Oertel, Joachim; Sattel, Thomas; Flaßkamp, Kathrin; Worthmann, Karl
Optimal path planning for stereotactic neurosurgery based on an elastostatic cannula model. - In: IFAC-PapersOnLine, ISSN 2405-8963, Bd. 55 (2022), 20, S. 600-605

In this paper, we propose a path-planning problem for stereotactic neurosurgery using concentric tube robots. The main goal is to reach a given region of interest inside the brain, e.g. a tumor, starting from a feasible point on the skull with an ideally short path avoiding certain sensitive brain areas. To describe the shape of the entire cannula from an entry point to the point of interest we use an existing mechanical model for continuum robots. We show numerically that our approach enables the surgeon to reach areas within the brain that would be impossible with a straight cannula as it is currently state of the art.
Zimmermann, Armin; Hotz, Thomas; Hädicke, Volker; Friebe, Martin
Analysis of safety-critical cloud architectures with multi-trajectory simulation. - In: IEEE Xplore digital library, ISSN 2473-2001, (2022), insges. 7 S.

Dynamic safety-critical systems require model-based techniques and tools for their systems design. The paper presents a stochastic Petri net model of an industrial safetycritical cloud server architecture for train control. Its reliability has to be evaluated to assess tradeoffs in architecture and level of fault tolerance. Simulation methods are too slow for such rare-event problems, while numerical analysis techniques suffer from the state-space explosion problem. The paper extends a recently developed multi-trajectory simulation algorithm combining elements of simulation and numerical analysis such that it increases the accuracy of rare-event simulations within a given computation time budget. Simulation experiments have been carried out with a prototype tool.
Schaller, Manuel; Wilson, Mitsuru; Kleyman, Viktoria; Mordmüller, Mario; Brinkmann, Ralf; Müller, Matthias A.; Worthmann, Karl
Parameter estimation and model reduction for model predictive control in retinal laser treatment. - In: Control engineering practice, ISSN 1873-6939, Bd. 128 (2022), 105320

Laser photocoagulation is one of the most frequently used treatment approaches for retinal diseases such as diabetic retinopathy and macular edema. The use of model-based control, such as Model Predictive Control (MPC), enhances a safe and effective treatment by guaranteeing temperature bounds. In general, real-time requirements for model-based control designs are not met since the temperature distribution in the eye fundus is governed by a heat equation with a nonlinear parameter dependency. This issue is circumvented by representing the model by a lower-dimensional system which well-approximates the original model, including the parametric dependency. We combine a global-basis approach with the discrete empirical interpolation method, tailor its hyperparameters to laser photocoagulation, and show its superiority in comparison to a recently proposed method based on Taylor-series approximation. Its effectiveness is measured in computation time for MPC. We further present a case study to estimate the range of absorption parameters in porcine eyes, and by means of a theoretical and numerical sensitivity analysis we show that the sensitivity of the temperature increase is higher with respect to the absorption coefficient of the retinal pigment epithelium (RPE) than of the choroid’s.