Mathematisches Forschungskolloquium 2024
Das Forschungskolloquium betrachtet mathematische Themen tiefergreifend und differenziert. So soll ein Austausch über die mathematischen Spezialisierungen der verschiedenen Arbeitsgruppen am Institut und darüber hinaus gefördert werden. Außerdem sollen auch Studierende (hauptsächlich Master-Studierende) durch das Kolloquium die Gelegenheit erhalten, sich über spezifische Aspekte mathematischer Themen zu informieren.
- Hr. Dr. Alexander Keimer (Friedrich-Alexander-Universität Erlangen, Department Mathematik )
"Recent results on nonlocal balance laws and the singular limit"
We will motivate the applicability of nonlocal conservation laws and then focus on classical existence and uniqueness results where the velocity field depends nonlocally on the solution, i.e., on the integral of the solution around a given spatial neighborhood. Interestingly, such problems can be treated by means of a fixed-point argument, avoiding the need of the famous entropy conditions which is crucial for classical (local) conservation laws.
We then discuss the related singular limit problem: If the kernel of the nonlocal term converges to a Dirac distribution, we formally obtain a classical (local) conservation law, but do the solutions of the nonlocal conservation law converge to the corresponding entropy solution of the local conservation law? Yes, under rather general and not very restrictive assumptions on the data which enables it to interprete local equations as limit of nonlocal equations.
We conclude with some open problems and further research directions.
05. August 2024, 14:15 Uhr, Raum HS 125 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Thomas Lorenz
- Hr. Hector Vargas Alvarez (Scuola Superiore Meridionale, Napoli/Italien - Modelling Engineering Risk and Complexity)
"Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning"
We present a physics-informed machine learning (PIML) scheme for the feedback linearization of nonlinear discrete-time dynamical systems. The PIML finds the nonlinear transformation law, thus ensuring stability via pole placement, in one step. In order to facilitate convergence in the presence of steep gradients in the nonlinear transformation law, we address a greedy training procedure. We assess the performance of the proposed PIML approach via a benchmark nonlinear discrete map for which the feedback linearization transformation law can be derived analytically; the example is characterized by steep gradients, due to the presence of singularities, in the domain of interest.
We show that the proposed PIML outperforms, in terms of numerical approximation accuracy, the "traditional'' numerical implementation, which involves the construction --and the solution in terms of the coefficients of a power-series expansion--of a system of homological equations as well as the implementation of the PIML in the entire domain, thus highlighting the importance of continuation techniques in the training procedure of PIML schemes.
We use physics-informed neural networks (PINNs) to numerically solve the discrete-time nonlinear observer-based state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation operator by solving a system of functional equations. The performance of the proposed approach is assessed via two illustrative case studies, for which the observer linearizing transformation operator can be derived analytically. We also perform an uncertainty quantification analysis for the proposed scheme, and we compare it with conventional power-series numerical implementation.
12. Juni 2024, 16:00 Uhr, Raum HS 326/327 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Jens Starke
- Prof. Dr. Hajo Holzmann (Philipps-Universität Marburg)
"Optimal rates of convergencefor estimating the mean function and the covariance kernel
in functional data analysis"
We derive optimal rates of convergence in the supremum norm for estimating the Hölder-smooth mean function as well as the covariance kernel of a stochastic process which is repeatedly and discretely observed with additional errors at fixed, synchronous design points, the typical scenario for machine recorded functional data. Similarly to the optimal rates in L_2 for the mean function obtained in Cai and Yuan (2011), for sparse design a discretization term dominates, while in the dense case the parametric root-n rate can be achieved as if the n-processes were continuously observed without errors. The supremum norm is of practical interest since it corresponds to the visualization of the estimation error, and forms the basis for the construction uniform confidence bands. We show that in contrast to the analysis in L_2, there is an intermediate regime between the sparse and dense cases dominated by the contribution of the observation errors. Furthermore, under the supremum norm interpolation estimators for the mean which suffice in L_2 turn out to be sub-optimal in the dense setting, which helps to explain their poor empirical performance. For the covariance kernel we devise estimators which make use of higher-order smoothness away from the diagonal without requiring the same smoothness on the diagonal, and thus are able to cover processes with relatively rough sample paths. We also obtain a central limit theorems in the supremum norm, and provide simulations and real data applications to illustrate our results.
12. Juni 2024, 15:00 Uhr, Raum SR 228 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Alexander Meister
- Paul-Erik Haacker (Universität Stuttgart, Institut für Nichtlineare Mechanik)
"Towards Stability Analysis of Fractional Differential Equations"
In the world of fractional differential equations (FDEs), one finds rich theories on unique phenomena but also scattered communities promoting competing philosophies. This talk will give a perspective on standing results, their limitations and fundamental decisions to make when working with FDEs. We aim to develop a numerical method of Hill-type to analyze stability properties of such systems.
05. Juni 2024, 15:15 Uhr, Raum HS 326/327 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Jens Starke
- Dr. Charlene Weiß (Universität Paderborn, Diskrete Mathematik)
"Codes and Designs"
Codes und Designs sind wichtige Objekte der Kombinatorik, die eng miteinander verbunden sind. Viele bekannte kombinatorische Probleme, die oft Anwendung in anderen Bereichen wie z. B. der Informationstheorie, der Geometrie oder der Quantenphysik haben, hängen mit der Bestimmung von großen Codes oder kleinen Designs zusammen. Ein mächtiger Ansatz zur Untersuchung dieser Objekte ist die Delsarte-Theorie, in der Codes und Designs als Teilmengen von Assoziationsschemata untersucht werden.
Wir betrachten zunächst Assoziationsschemata und dann Delsartes bahnbrechende Methode der linearen Optimierung, die obere Schranken für Codes liefert. Anschließend werden wir uns auf Codes und Designs in Polarräumen konzentrieren, die aus total isotropen Unterräumen eines endlichen Vektorraums mit einer nicht-entarteten Form bestehen. Mithilfe Delsartes Methode werden wir Schranken für Codes in Polarräumen herleiten und mit diesen eine fast vollständige Klassifizierung von Steiner-Systemen in Polarräumen geben.
22. Mai 2024, 15:00 Uhr, Raum SR 228 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan
- Dr. Daniel Hauer (BTU Cottbus)
"The fundamental gap conjecture"
18. April 2024, 13:00 Uhr, Raum SR120 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Peter Takác