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AG Modellierung, Numerik, DifferentialgleichungenKolloquium SS 2019

"AG Modellierung, Numerik, Differentialgleichungen"

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Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Sommersemester 2019
Verantwortliche Dozenten:
Alle Professoren der
Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen
Koordination:
Dr. Janusz Ginster
Termine:
Di 16-18 Uhr in MA 313 und nach Vereinbarung
Inhalt:
Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen
Kontakt:

Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Mitglieder der Arbeitsgruppe besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Terminplanung / schedule (Abstracts s. unten / Abstracts see below)
Datum
date
Zeit
time
Raum
room
Vortragende(r)
speaker
Titel
title
Einladender
invited by
09.04.2019
16:15
MA 004
Eduardo Casas Rentería (Universidad de Cantabria)
Colloquium in honor of the retirement of Prof. Dr. Fredi Tröltzsch
R. Schneider
16.04.2019
16:15
MA 313
23.04.2019
16:15
MA 313

 
30.04.2019
16:15
MA 313
 
07.05.2019
16:15
MA 313

14.05.2019
16:15
MA 313
21.05.2019
16:15
MA 313
28.05.2019
16:15
MA 313
Gitta Kutyniok (TU Berlin)
B. Zwicknagl
04.06.2019
16:15
MA 313
11.06.2019
16:15
MA 313
Christopher Beattie (Virginia Tech)
V. Mehrmann
19.06.2019 (Wednesday)
16:15
MA 313
Serkan Gugercin (Virginia Tech)
V. Mehrmann
25.06.2019
16:15
MA 313
02.07.2019
16:15
MA 313
09.07.2019
16:15
MA 313
Helmut Harbrecht (Universität Basel)
R. Schneider

Abstracts zu den Vorträgen

Eduardo Casas Rentería (Universidad de Cantabria):  Optimality Conditions in Control of Partial Differential Equations 
In this talk, I will consider the first and second order optimality conditions for control problems governed by partial differential equations. The first order conditions provide some qualitative properties of the optimal controls. They are also crucial for the numerical computation of the solutions. The second order conditions allow to prove the stability of the solutions with respect to small perturbations of the data and to derive error estimates for the numerical discretization of the control problem. Moreover, they play an important role in the analysis of the convergence order of the computational methods. I will emphasize the difference between the sufficient second order conditions for finite and infinite optimization problems. I will present some well established results along with some new ones recently proved.

Gitta Kutyniok (TU Berlin):  Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs 
High-dimensional parametric partial differential equations (PDEs) appear in various contexts including control and optimization problems, inverse problems, risk assessment, and uncertainty quantification. In most such scenarios the set of all admissible solutions associated with the parameter space is inherently low dimensional. This fact forms the foundation for the so-called reduced basis method.
Recently, numerical experiments demonstrated the remarkable efficiency of using deep neural networks to solve parametric problems. In this talk, we will present a theoretical justification for this class of approaches. More precisely, we will derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric PDEs. In fact, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical approximation results. We use this low-dimensionality to guarantee the existence of a reduced basis. Then, for a large variety of parametric PDEs, we construct neural networks that yield approximations of the parametric maps not suffering from a curse of dimensionality and essentially only depending on the size of the reduced basis. This is joint work with Philipp Petersen (Oxford), Mones Raslan, and Reinhold Schneider.

Christopher Beattie (Virginia Tech):  Data-driven identification of dissipative dynamics  
System identification has evolved toward a methodology for the construction of approximate dynamic models based on observations of system behavior possibly conjoined with structural constraints that the true system is expected to respect. For example, computational models of physical systems should take into account the manner in which systems handle energy flux and more broadly, conservation laws, but this can be a significant challenge when models are derived directly from system response data; observational noise can further complicate the enterprise.
I will discuss a few frameworks for considering energy conservation and dissipation for dynamical systems and describe how one can ascertain whether an observed response profile is compatible with a particular dissipation/conservation model. This leads naturally to a data-driven modeling framework that has features in common with classic Nevanlinna-Pick interpolation and port-Hamiltonian modeling. The final product is a convex parameterized family of dissipative models all of which are consistent with observed response profiles and well-suited for further use in design and control applications.

Serkan Gugercin (Virginia Tech):  Data-driven modeling for solving nonlinear eigenvalue problems and estimating dispersion curves  
Projection-based methods are a common approach to model reduction in which reduced-order quantities are obtained via explicit use of full-order quantities. However, these full-order quantities are not always accessible and instead a large-set of input/output data, e.g., in the form of transfer function evaluations, are available. In this talk, we will focus on both interpolation (Loewner) and least-squares (Vector Fitting) frameworks to construct reduced-models directly from data. In the former case, we will connect the data-driven modeling framework to nonlinear eigenvalue problems and discuss how the classical realization theory gives further insights into certain classes of methods for nonlinear eigenvalue computations. In the Vector Fitting framework, we will show how one can use data-driven modeling in estimating dispersion curves in structural materials.

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