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Summer Term 2019

Mar 21, 2019 - Jun 27, 2019

Hosts: Prof. Dr. R. Klein (FU), Prof. Dr. R. Kornhuber (FU), Prof. Dr. C. Schütte (FU/ZIB)
Location: Freie Universität Berlin, Institut für Mathematik, Arnimallee 6, 14195 Berlin-Dahlem, Room: 032 ground floor
Time: The seminar takes place on Thursday at 4:00 pm

Thursday, 21.03.2019: Lecture

Oliver Bühler, New York University
The Kolmogorov code of turbulence and GFD

Fluid turbulence is the quintessential example of a nonlinear multiscale process and in its rotating and stratified forms it is of prime importance in GFD. In this talk we consider how some exact results of classical turbulence theory have recently been adapted for use in GFD. This centers on third-order structure functions and their use in determining spectral energy fluxes from in situ measurements in the atmosphere and ocean. I will explain the relevant classical theory and its adaptation, and will give examples of idealized fluid flows, including a form of 2d magnetohydrodynamic turbulence.

Friday, 05.04.2019: Joint Colloquium of CRC 1114 and CRC 1294

University of Potsdam
Campus Griebnitzsee
Building 06, Lecture Hall 3
August-Bebel-Str. 89
14482 Potsdam    

Felix Otto, MPI für Mathematik in den Naturwissenschaften, Leipzig
Effective behavior of random media

An introduction lecture for PhD students and postdocs will be given by Nikolas Nüsken (Universität Potsdam) from 9:00am to 9:45am in the same lecture hall.

Felix Otto will speak as invited guest during this joint colloquium of SFB 1114 and SFB 1294 on the Effective Behavior of random media Abstract: In engineering applications, heterogeneous media are often described in statistical terms. This partial knowledge is sucient to determine the effective, i. e. large-scale behavior. This effective behavior may be inferred from the Representative Volume Element (RVE) method. I report on last years' progress on the quantitative understanding of what is called stochastic homogenization of elliptic partial differential equations: optimal error estimates of the RVE method, leading-order characterization of fluctuations, effective multipole expansions. Methods connect to elliptic regularity theory and to concentration of measure arguments.

Thursday, 11.04.2019: Colloquium

Robert Polzin, Freie Universität Berlin:
"What is... DBMR?"

In this talk, a recent method of Susanne Gerber and CRC 1114 Mercator fellow Illia Horenko, DBMR, is discussed. The method constructs a directly low-rank transfer operator, reducing numerical effort and error due to finite data. Given two categorical random variables with respective ranges, the aim is to find a stochastic matrix of conditional probabilities between the discrete states of both processes. The usual maximum-likelihood estimate of the matrix requires a large amount of pair observations. Gerber and Horenko suggest an efficient and scalable estimation of the transfer operator by introducing intermediate latent states.

Thursday, 25.04.2019: Colloquium

Martin Heida, WIAS, & Marcus Weber, ZIB:
"What is... SQRA discretization of the Fokker-Planck equation?"

Estimating the probability of rare events with the help of sampling methods is a difficult problem area of our CRC 1114. Particularly in the field of molecular simulations, it is important to know the rate at which chemical complexes are formed or dissociate. In SQRA (square root approximation), these type of processes are discretized by a finite volume operator in the state space where the transition rates between "neighboring" states of the system are estimated in a special mathematical way. The resulting SQRA operator has good qualitative (high dimension) and quantitative (low dimension) convergence properties. It is related to the known exponential fitting scheme (Scharfetter-Gummel) but imposes less difficulties in the proof of convergence. However, we can show that these two methods are asymptotically equivalent.

Thursday, 09.05.2019: Lecture

Claude Bardos, Laboratoire Jacques Louis Lions - Paris 6

From the d'Alembert paradox to the 1984 Kato criteria via the 1941  $1/3$ Kolmogorov law and the 1949 Onsager conjecture
Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann , Piotr and Agneska Gwiadza , were motivated by the following issues:
The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence.
As a consequence.
I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is equivalent to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.

Thursday, 16.05.2019: Lecture

Ralf Metzler, Universität Potsdam
Brownian Motion and Beyond

Roughly 190 years ago Robert Brown reported the "rapid oscillatory motion" of microscopic particles, the first systematic study of what we now call Brownian motion. At the beginning of the 20th century Albert Einstein, Marian Smoluchowski, and Pierre Langevin formulated the mathematical laws of diffusion. Jean Perrin's experiments 110 years ago then prompted a very active field of ever refined diffusion experiments.
Despite the long-standing history of Brownian motion, after an historic introduction I will report several new developments in the field of diffusion and stochastic processes. This new research has been fuelled mainly by novel insights into complex microscopic systems such as living biological cells, made possible by Nobel-Prize winning techniques in laser physics, superresolution microscopy, or through supercomputing studies. Topics covered include Brownian yet non-Gaussian diffusion, the geometry-control of first passage statistics, and anomalous diffusion with a power-law time dependence of the mean squared displacement. For the latter, questions of ergodicity and ageing will be discussed.

Thursday, 13.06.2019: Lecture

Peter Friz, TU Berlin
Multiscale Systems, Homogenization and Rough Paths

Rough paths (and its recent generalisations: paracontrolled calculus, regularity structures) provide a powerful framework to the analysis of (partial) differential equations equations. Typical applications include highly-oscillatory systems (a priori well-posed, but with unclear limiting behaviour) and stochastic equations (analytically ill-posed, though sometimes within reach of Ito's stochastic analysis). After presenting some general ideas, I will explain how rough paths have been used to solve a concrete fast-slow homogenisation problem originally posed in [5].
[1] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Deterministic homogenization for discrete-time fast-slow systems under optimal moment assumptions. arXiv 2019
[2] Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang; Multiscale systems, homogenization, and rough paths; arXiv 2017 and 2019 Springer Volume Varadhan 75
[3] Peter Friz, Paul Gassiat and Terry Lyons, Physical Brownian motion in a magnetic field as a rough path. Trans. AMS 2015
[4] David Kelly, Ian Melbourne, Deterministic homogenization for fast-slow systems with chaotic noise, Journal of Functional Analysis 2017
[5] Melbourne and A. M. Stuart; Diffusion limits of chaotic skew-product flows, Nonlinearity 2011.

Thursday, 20.06.2019: Lecture

Upanshu Sharma, École des Ponts Paris Tech
Estimating coarse-graining error beyond reversibility

Coarse-graining is the procedure of approximating a complex and high-dimensional system by a simpler and lower dimensional one. Typically, such an approximation is achieved by using a coarse-graining map  F, which projects the full state of a system  X  (representing for instance the position of particles in the system), onto a lower-dimensional state space. Assuming that the state X evolves according to a stochastic differential equation (SDE), it is easy to identify the evolution of  F(X)  — however, this cannot be used in practice since the evolution still depends on the original state space. Legoll and Lelièvre (2010) addressed this by introducing a natural approximation, called the effective dynamics, and quantified the error of this approximation when  X  solves the overdamped Langevin equation. Since then a variety of results have been discussed in this direction, in the setting when  X  evolves according to a reversible SDE. In this talk, I will show that reversibility is not a limitation, and both the construction of the effective dynamics and the error estimates can be generalised to the setting of non-reversible SDEs. This is joint work with C. Hartmann and L. Neureither.

Thursday, 27.06.2019: Colloquium

Abhishek Harikrishnan, FU Berlin: "What is... a shearlet?"

Shortcomings in Fourier or Wavelet representations of multivariate data, most commonly images, have lead to the development of shearlets, originally introduced for the sparse approximation of functions from  L2(R2).  They are a multiscale geometric framework tuned to efficiently encode anisotropic features of such functions using parabolic scalings and shearings of some mother shearlet psi. In this talk, I will show the construction of shearlet frames, as well as a theorem showing almost-optimal sparse approximation behavior of shearlets with respect to a model class of images with anisotropic features (so-called cartoon-like functions). Moreover, I will present a few examples of applications of shearlets from image processing (denoising, inpainting, super-resolution, medical imaging...) by means of sparse regularization, which has been proven to be a useful prior in such imaging problems.

Johannes von Lindheim, TU Berlin: "What is... a coherent structure?"

Although a formal definition of a coherent (or turbulent) structure has so far been elusive, it is generally thought to be a region of space and time within which the flow field exhibits a characteristic coherent pattern. With the contribution from flow visualization experiments and DNS simulations, Pope (2001) classifies these structures into eight categories. In this talk, I will give a brief overview on turbulent structures and then shift the focus to identifying one category of such structures, namely vortices, both in the Eulerian and Lagrangian specification of the flow field.