Events & Teaching

Upcoming Events

go to archive

Tuesday, 30.08.2016, 15.00 Uhr (WIAS-406)
Seminar Partielle Differentialgleichungen
M. Varga, Technische Universität Dresden:
Homogenization via stochastic unfolding
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
Two-scale convergence methods have been proven to be convenient tools in homogenization of various equations. Problems with rapidly oscillating periodic coefficients are treated using the notion of two-scale convergence. The development of the periodic unfolding operator facilitated a simpler two-scale analysis of such problems by embedding sequences of oscillating functions into an appropriate two-scale space. The notion of stochastic two-scale convergence in the mean has been used to obtain homogenization results for problems involving oscillating random and stationary coefficients. We adapt this notion to the setting of functions defined on a discrete physical space by introducing an appropriate stochastic unfolding operator. We investigate the properties of this operator which provides a basis for a simple method for convex homogenization.

Further Informations
Seminar Partielle Differentialgleichungen

Host
WIAS Berlin
September 8 – 9, 2016 (WIAS-ESH)
Workshop/Konferenz: 1st Leibniz MMS Mini Workshop on CFD & GFD
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Host
WIAS Berlin
September 12 – 16, 2016 (WIAS-ESH)
Workshop/Konferenz: WIAS/GAMM/TUM Summer School on Uncertainty Quantification 2016
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstract
https://www.wias-berlin.de/workshops/uq-school16/

Host
GAMM (AG UQ)
Technische Universität München
WIAS Berlin
Wednesday, 14.09.2016, 15.15 Uhr (WIAS-405-406)
BERLINER OBERSEMINAR Nichtlineare partielle Differentialgleichungen (Langenbach-Seminar)
Prof. J. Feng, University of Kansas, USA:
Hamilton--Jacobi PDEs in the space of probability measures, the metric nature explored
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406

Abstract
Hamilton-Jacobi PDE in the space of probability measure is a new class of PDE. We give two examples. The first example concerns a statistical mechanics application where the equation is derived by probabilists studying Gibbs-Non-Gibbs transitions in the Netherlands community. Through a method by the speaker and Tom Kurtz, the equation's uniqueness will rigorously give the large deviation principle. However, this uniqueness problem is still open. This problem has a hidden spatial translation invariance. The second example concerns the variational formulation of a compressible Euler equation in continuum mechanics. The probability measure is just the density profile of infinite particles. The Hamilton-Jacobi equation characterizes the canonical transformations and is expected to give useful information on the large time dynamics of the Euler-Lagrange equation (Aubry-Mather theory for instance). We have now a well-podesness theory for this equation. This example has a hidden particle permutation invariance in the density profile representation In the rigorous part of this talk, I focus on explaining how the well-posedness in the mechanical application is solved. An important observation is that the space of probability measure is best viewed as an infinite dimensional quotient space in this context. The quotient structure comes from particle permutation symmetry. To treat a PDE in such an infinitely folded space, we devise techniques based on the metric space analysis and the Wasserstein spaces. A key step is the use of geometric tangent cone concept in characterizing the velocity variables. Admissible velocity fields are more than function valued, they are relaxed to belong to a subspace of the space of Markov transition kernels. This augmentation of the tangent space is essential to allow us distinguish curves with mass condensation property in the physical space density profile variable with other curves without such feature in the Hamiltonian formulation. The infinitely folded space structure here also has an intrinsic connection with the space of random variables. In studying the mean-field game theory, P.L. Lions setup some Hamilton-Jacobi equation in the space of random variables. A connection between the measure formulation here and the random-variable formulation there are conceptually possible but many questions are still left open.

Further Informations
Berliner Oberseminar ``Nichtlineare Partielle Differentialgleichungen'' (Langenbach Seminar)

Host
WIAS Berlin
Humboldt-Universität zu Berlin
Thursday, 22.09.2016, 16.00 Uhr (WIAS-ESH)
Forschungsseminar Mathematische Modelle der Photonik
Dr. O. Chvartatskyi, Max-Planck Institute for Dynamics and Self-Organisation:
NLS breathers, rogue waves, and solutions of the Lyapunov equation
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Host
WIAS Berlin