Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations - Cresson, Jacky; Jiménez, Fernando; Ober-Blöbaum, Sina (AIMS)

Turnpike Property in Optimal Microbial Metabolite Production - Caillau, Jean-Baptiste; Djema, Walid; Gouzé, Jean-Luc; Maslovskaya, Sofya; Pomet, Jean-Baptiste (Journal of Optimization Theory and Applications)
Abstract We consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). According to Pontryagin’s maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or Fuller phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the turnpike property. This property is further used in order to design simple and realistic suboptimal control strategies.
Keywords: Applied Mathematics; Management Science and Operations Research; Control and Optimization

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds - Cekić, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P. (Inventiones mathematicae)
Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$Σ $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$SΣ $$ S Σ with harmonic 1-forms on $$Σ $$ Σ .
Keywords: General Mathematics

Symmetrieverständnis von Anfang an - Häsel-Weide, Uta; Wallner, Melina; Hattermann, M. (Anfangsunterricht für alle Kinder - Willkommen in der Schule!)

Professionalisierung für inklusiven Mathematikunterricht. Interdisziplinäre Seminarkonzeption zur reflexiven Professionalisierung angehender Mathematiklehrkräfte in der Sekundarstufe - Häsel-Weide, Uta; Seitz, S.; Wallner, Melina; Wilke, Y. (Qualifizierung für Inklusion. Sekundarstufe)

Diagnosegeleitete Förderung im inklusiven Mathematikunterricht der Grundschule – Professionalisierung durch reflektierte Handlungspraxis in der Lehrer*innenbildung. - Hähn, K.; Häsel-Weide, Uta; Scherer, P. (QfI - Qualifizierung für Inklusion)

Koopman analysis of quantum systems - Klus, Stefan; Nüske, Feliks; Peitz, Sebastian (Journal of Physics A: Mathematical and Theoretical)
Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science.

Using second-order information in gradient sampling methods for nonsmooth optimization - Gebken, Bennet (arXiv:2210.04579)
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.

Non-Lie subgroups in Lie groups over local fields of positive characteristic - Glöckner, Helge (p-Adic Numbers, Ultrametric Analysis, and Applications)
Keywords: 20Exx; 22Exx; 32Cxx

Manifolds of mappings on Cartesian products - Glöckner, Helge; Schmeding, Alexander (Annals of Global Analysis and Geometry)
Keywords: 58D15; 22E65; 26E15; 26E20; 46E40; 46T20; 58A05

Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces - Glöckner, Helge (Axioms)
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.

Birkhoff decompositions for loop groups with coefficient algebras - Glöckner, Helge (arXiv:2206.11711)
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.

Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization - Gebken, Bennet
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.

Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients - Berkemeier, Manuel Bastian; Peitz, Sebastian (arXiv:2208.12094)
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.

On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation - Bieker, Katharina; Gebken, Bennet; Peitz, Sebastian (IEEE Transactions on Pattern Analysis and Machine Intelligence)
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.

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature - Kolb, Martin; Weich, Tobias; Wolf, Lasse Lennart (Annales Henri Poincaré)
Abstract The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$ M is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$ M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$ L 2 -spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Ruelle–Pollicott resonances for manifolds with hyperbolic cusps - Guedes Bonthonneau, Yannick; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data - Hesse, Kerstin; Le Gia, Quoc Thong (Journal of Computational and Applied Mathematics)
Keywords: Applied Mathematics; Computational Mathematics

Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian