Martin Heida, WIAS Berlin, & Marcus Weber, Zuse Institut Berlin

"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.

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.

Oliver Bühler, Courant Institute of Mathematical Sciences, NYU

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.

Felix Otto, MPI für Mathematik in den Naturwissenschaften, Leipzig

Effective behavior of random media

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.

Ralf Metzler, Universität Potsdam, Institut für Physik und Astronomie

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.

Back to top