19.09.2016

Dirichlet-to-Neumann maps and the spectral data of Schrödinger operators, Jussi Behrndt, TU Graz

14:00, Reichenhainer Str. 41, Room 705

14.09.2016

Brownian Motion on Manifolds and Gradient Estimates via the Bismut-Elworthy-Li Formula, Robert Baumgarth,TU Dresden

In the first part of the talk we recall well known results [Batty-Duyckaerts 08, Borichev-Tomilov 10] which permit to characterize the rate of energy decay via the rate at which the corresponding resolvent operator growth along the imaginary axis. We give an equivalent characterization of the resolvent estimates in terms of a handy inequality. In the second part we discuss the damped wave equation on a 2D partially rectangular domain with a strongly trapped geodesic. I present known results and open questions for this case. The third part of the talk is devoted to a sketch of the proof of the aforementioned handy inequality.

15:30, Reichenheiner Straße 41, Raum 705

Abstract: A System of differential equations arising in the theory of nonlinear oscillations in resonance-related problems is considered. Of special interest are solutions whose amplitude increases with time. Such solutions correspond to autoresonance phenomenon. Explicit formulas for solutions can't be obtained. However, it is possible to construct an asymptotic expansion for some particular solutions. The stability of such type solutions with respect to random perturbations is analysed.

15:30, Reichenheiner Straße 41, Raum 705

Abstract: Dies ist ein einführender Vortrag, der einige Grenzverteilungssätze, die bei der stochastischen Modellierung komplexer Systeme Verwendung finden, vorstellt. Dabei ist der Begriff Grenzverteilungssatz weit gefasst und umfasst Sätze der großen Zahlen bzw. Ergodensätze, Limiten von Punktprozessen und die Charakterisierung von asymptotischen Eigenschaften von Verteilungen, die als Grenzwert auftreten. Die Grenzwertsätze ermöglichen es, Funktionen von unendlich vielen Zufallsvariablen durch solche von endlich vielen zu approximieren. Es werden Beispiele aus der Perkolationstheorie und von Eigenwertstatistiken von zufälligen Hamilton-Operatoren vorgestellt, in denen diese Konvergenzsätze Anwendung finden.

Uncertainty quantification, Virginie Ehrlacher, Ecole des Ponts ParisTech, France

10:30, Reichenheiner Straße 41, Raum 705

Abstract: In this talk, we will present a basic introduction to uncertainty quantification methods. The aim of uncertainty quantification is the following: given the solution of a PDE with stochastic coefficients, we would like to compute output quantities like statistical averages or failure probability knowing the law of input quantities. After the description of the well-known Karhunen-Loeve decomposition, which is the basic mathematical tool used in order to represent a square-integrable processes by means of a finite number of random variables, we will present the most classical stochastic and deterministic techniques used in uncertainty quantification in order to compute means, failure probabilities and probability distribution of output quantities, like Monte-Carlo, reliability methods, Neumann, perturbation, L2 projection, collocation and Galerkin methods. If there is enough time, a brief introduction to methods specifically designed to circumvent the curse of dimensionality will be presented.

Periodic Schrödinger operators with local defects and spectral pollution, Virginie Ehrlacher, Ecole des Ponts ParisTech, France

15:30, Reichenheiner Straße 41, Raum 705

Abstract: This talk will deal with the numerical calculation of eigenvalues of perturbed periodic Schrödinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. We show that the corresponding spurious eigenvectors can be interpreted as surface states in a certain sense which will be precised. Besides, we extend results by Lewin and Séré on some no-pollution criteria and prove that spectral pollution can be avoided using augmented finite element bases using the so-called Wannier functions. Indeed, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Schrödinger operator. Lastly, we prove that the supercell method does not lead to spectral pollution and provide a priori estimates for this approximation method. It is proved in particular that the rate of convergence of the method is exponential with respect to the size of the supercell.

Spectral averaging for random operators with a summability condition on singular values, Matthias Täufer,

29.01.2014

Pointwise ergodic theorems for the free group - the Bowen/Nevo approach, Felix Pogorzelski, Friedrich-Schiller-Universität Jena,

13:30, Reichenheiner Straße 41, Raum 705

Abstract: The theory of the dynamics of measure preserving group actions on probability spaces is a key discipline in ergodic theory. During the last couple of years,significant breakthroughs have been made concerning the pointwise convergence theorems for actions on non-amenable groups. In particular, Lewis Bowen and Amos Nevo have developed a strong machinery to prove ergodic theorems for large classes of group. The goal of the talk is to give a brief description of the method in the case of the free group. We will point out how to find amenable equivalence relations on the boundary of the group which can be used to use classical tools for beautiful non-classical results.