@unpublished{34618, abstract = {{In this article, we show how second-order derivative information can be incorporated into gradient sampling methods for nonsmooth optimization. The second-order information we consider is essentially the set of coefficients of all second-order Taylor expansions of the objective in a closed ball around a given point. Based on this concept, we define a model of the objective as the maximum of these Taylor expansions. Iteratively minimizing this model (constrained to the closed ball) results in a simple descent method, for which we prove convergence to minimal points in case the objective is convex. To obtain an implementable method, we construct an approximation scheme for the second-order information based on sampling objective values, gradients and Hessian matrices at finitely many points. Using a set of test problems, we compare the resulting method to five other available solvers. Considering the number of function evaluations, the results suggest that the method we propose is superior to the standard gradient sampling method, and competitive compared to other methods.}}, author = {{Gebken, Bennet}}, booktitle = {{arXiv:2210.04579}}, title = {{{Using second-order information in gradient sampling methods for nonsmooth optimization}}}, year = {{2022}}, } @article{34792, author = {{Glöckner, Helge}}, issn = {{2070-0466}}, journal = {{p-Adic Numbers, Ultrametric Analysis, and Applications}}, keywords = {{20Exx, 22Exx, 32Cxx}}, number = {{2}}, pages = {{138–144}}, title = {{{Non-Lie subgroups in Lie groups over local fields of positive characteristic}}}, doi = {{10.1134/S2070046622020042}}, volume = {{14}}, year = {{2022}}, } @article{34791, author = {{Glöckner, Helge and Schmeding, Alexander}}, issn = {{0232-704X}}, journal = {{Annals of Global Analysis and Geometry}}, keywords = {{58D15, 22E65, 26E15, 26E20, 46E40, 46T20, 58A05}}, number = {{2}}, pages = {{359–398}}, title = {{{Manifolds of mappings on Cartesian products}}}, doi = {{10.1007/s10455-021-09816-y}}, volume = {{61}}, year = {{2022}}, } @article{34796, abstract = {{We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.}}, author = {{Glöckner, Helge}}, issn = {{2075-1680}}, journal = {{Axioms}}, number = {{5}}, title = {{{Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces}}}, doi = {{10.3390/axioms11050221}}, volume = {{11}}, year = {{2022}}, } @unpublished{34804, abstract = {{Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with valued in complex Banach-Lie groups.}}, author = {{Glöckner, Helge}}, booktitle = {{arXiv:2206.11711}}, title = {{{Birkhoff decompositions for loop groups with coefficient algebras}}}, year = {{2022}}, } @phdthesis{31556, abstract = {{Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind.}}, author = {{Gebken, Bennet}}, title = {{{Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization}}}, doi = {{10.17619/UNIPB/1-1327}}, year = {{2022}}, } @unpublished{33150, abstract = {{In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem.}}, author = {{Berkemeier, Manuel Bastian and Peitz, Sebastian}}, booktitle = {{arXiv:2208.12094}}, title = {{{Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients}}}, year = {{2022}}, } @article{20731, abstract = {{We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.}}, author = {{Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian}}, journal = {{IEEE Transactions on Pattern Analysis and Machine Intelligence}}, number = {{11}}, pages = {{7797--7808}}, publisher = {{IEEE}}, title = {{{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}}}, doi = {{10.1109/TPAMI.2021.3114962}}, volume = {{44}}, year = {{2022}}, } @article{35306, author = {{Guedes Bonthonneau, Yannick and Weich, Tobias}}, issn = {{1435-9855}}, journal = {{Journal of the European Mathematical Society}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{3}}, pages = {{851--923}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Ruelle–Pollicott resonances for manifolds with hyperbolic cusps}}}, doi = {{10.4171/jems/1103}}, volume = {{24}}, year = {{2022}}, } @article{34633, author = {{Hesse, Kerstin and Le Gia, Quoc Thong}}, issn = {{0377-0427}}, journal = {{Journal of Computational and Applied Mathematics}}, keywords = {{Applied Mathematics, Computational Mathematics}}, publisher = {{Elsevier BV}}, title = {{{L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data}}}, doi = {{10.1016/j.cam.2022.114118}}, volume = {{408}}, year = {{2022}}, } @article{34817, author = {{Hanusch, Maximilian}}, issn = {{1019-8385}}, journal = {{Communications in Analysis and Geometry}}, keywords = {{regularity of Lie groups}}, number = {{1}}, pages = {{53--152}}, publisher = {{International Press of Boston}}, title = {{{Regularity of Lie groups}}}, doi = {{10.4310/cag.2022.v30.n1.a2}}, volume = {{30}}, year = {{2022}}, } @techreport{34856, author = {{Hanusch, Maximilian}}, pages = {{385}}, publisher = {{https://maximilianhanusch.wixsite.com/my-site/lehre-teaching}}, title = {{{Analysis 1 und 2 Skript/Buch}}}, year = {{2022}}, } @article{35644, author = {{Kolb, Martin and Klump, Alexander}}, journal = {{Theory of Probability and its Applications}}, number = {{4}}, pages = {{717--744}}, publisher = {{Society for Industrial and Applied Mathematics}}, title = {{{Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary}}}, volume = {{67}}, year = {{2022}}, } @article{35649, abstract = {{Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.}}, author = {{Kolb, Martin and Liesenfeld, Matthias}}, journal = {{Electronic Journal of Probability}}, number = {{27}}, pages = {{1--28}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{On non-extinction in a Fleming-Viot-type particle model with Bessel drift}}}, doi = {{https://doi.org/10.1214/22-EJP866}}, year = {{2022}}, } @article{35650, abstract = {{We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn} are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.}}, author = {{Denisov, Denis and Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}}, journal = {{Electronic Journal of Probability}}, pages = {{1--43}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{Persistence of autoregressive sequences with logarithmic tails}}}, doi = {{https://doi.org/10.48550/arXiv.2203.14772}}, volume = {{27}}, year = {{2022}}, } @inproceedings{31367, author = {{Hoffmann, Max and Biehler, Rolf}}, booktitle = {{ Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen }}, editor = {{Halverscheid, Stefan and Kersten, Ina and Schmidt-Thieme, Barbara}}, pages = {{351--368}}, publisher = {{Springer Spektrum}}, title = {{{Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie.}}}, year = {{2022}}, } @inproceedings{31365, author = {{Biehler, Rolf and Hoffmann, Max}}, booktitle = {{ Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik}}, editor = {{Isaev, Viktor and Eichler, Andreas and Loose, Frank}}, isbn = {{9783662639474}}, issn = {{2197-8751}}, pages = {{ 49–72}}, publisher = {{Springer}}, title = {{{Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium}}}, doi = {{10.1007/978-3-662-63948-1_4}}, year = {{2022}}, } @inbook{45373, author = {{Dröse, Jennifer and Neugebauer, P. and Delucchi Danhier, R. and Mertins, B.}}, booktitle = {{Eye-Tracking in der Mathematik- und Naturwissenschaftsdidaktik. Forschung und Praxis}}, editor = {{Kleine, P. and Graulich, N. and Kuhn, J. and Schindler, M.}}, pages = {{209--225}}, title = {{{Eye-Tracking Studie zu Textaufgaben in Klasse 5: Bemerken und Interpretieren syntaktischer Strukturen}}}, doi = {{https://doi.org/10.1007/978-3-662-63214-7}}, year = {{2022}}, } @inproceedings{30733, abstract = {{Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{London}}, pages = {{1870--1875}}, title = {{{Verification of safety critical control policies using kernel methods}}}, year = {{2022}}, } @inproceedings{45378, author = {{Dröse, Jennifer and Wessel, Lena}}, booktitle = {{Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education. PME}}, editor = {{Fernandez, C. and Llinares, S. and Gutiérrez, A. and Planas, N.}}, title = {{{Prospective Teachers‘ Competence of Fostering Students’ Understanding in Script Writing Task}}}, year = {{2022}}, }