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.

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.

Helmut Harbrecht (Universität Basel):  Analytical and numerical methods in shape optimization  
Shape optimization is indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals which are de- fined over a class of admissible domains.
The application of gradient based minimization algorithms involves the shape functionals’ derivative with respect to the domain under consideration. Such derivatives can analytically be computed by means of shape calculus and enable the paradigm first optimize then discretize. Especially, by identifying the sought domain with a parametrization of its boundary, the solution of the shape optimization problem will be equivalent to solving a nonlinear pseudodifferential equation for the unknown parametrization.
The present talk aims at surveying on analytical and numerical methods for shape optimization. In particular, besides several applications of shape optimization, the following items will be addressed:
• first and second order optimality conditions
• discretization of shapes
• existence and convergence of approximate shapes
• efficient numerical techniques to compute the state equation

• Kolloquium ModNumDiff Winter 2018/2019 [2]
• Kolloquium ModNumDiff Sommer 2018 [3]
• Kolloquium ModNumDiff Winter 2017/2018 [4]
• Kolloquium ModNumDiff Sommer 2017 [5]
• Kolloquium ModNumDiff Winter 2016/17 [6]
• Kolloquium ModNumDiff Sommer 2016 [7]
• Kolloquium ModNumDiff Winter 2015/16 [8]
• Kolloquium ModNumDiff Sommer 2015 [9]
• Kolloquium ModNumDiff Winter 2014/15 [10]
• Kolloquium ModNumDiff Sommer 2014 [11]
• Kolloquium ModNumDiff Winter 2013/14 [12]
• Kolloquium ModNumDiff Sommer 2013 [13]
• Kolloquium ModNumDiff Winter 2012/13 [14]
• Kolloquium ModNumDiff Sommer 2012 [15]
• Kolloquium ModNumDiff Winter 2011/12 [16]
• Kolloquium ModNumDiff Sommer 2011 [17]