L3

There is a wide range of problems in continuum mechanics that involve transformation fronts, which are non-stationary interfaces between two different phases in a phase-transforming or a chemically-transforming material. From the mathematical point of view, the considered problems are represented by systems of non-linear PDEs with discontinuities across non-stationary interfaces, kinetics of which depend on the solution of the PDEs. Such problems have a significant industrial relevance – an example of a transformation front is the localised stress-affected chemical reaction in Li-ion batteries with Si-based anodes. Since the kinetics of the transformation fronts depends on the continuum fields, the transformation front propagation can be decelerated and even blocked by the mechanical stresses. This talk will focus on three topics: (1) the stability of the transformation fronts in the vicinity of the equilibrium position for the chemo-mechanical problem, (2) a fictitious-domain finite-element method (CutFEM) for solving non-linear PDEs with transformation fronts and (3) an applied problem of Si lithiation.

TBC

iii) identify a special class of correlated compressors based on the idea of random permutations, for which we coin the term PermK. The use of this technique results in the strict improvement on the previous MARINA rate. In the low Hessian variance regime, the improvement can be as large as √n, when d > n, and 1 + √d/n, when n<=d, where n is the number of workers and d is the number of parameters describing the model we are learning.

Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate probabilistic variant, the adapted Wasserstein distance $AW$, can play a similar role for the class $FP$ of filtered processes, i.e. stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob-decomposition, optimal stopping and stochastic control are continuous w.r.t. $AW$. We also show that $(FP, AW)$ is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace. Finally we consider computational aspects and provide a novel method based on the Sinkhorn algorithm.

The talk is based on articles with Daniel Bartl, Mathias Beiglböck and Stephan Eckstein.

I will present a simple abstract quantitative method for proving the hydrodynamic limit of interacting particle systems on a lattice, both in the hyperbolic and parabolic scaling. In the latter case, the convergence rate is uniform in time. This "consistency-stability" approach combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates à la Kruzhkov in weak distance, and consistency estimates exploiting the regularity of the limit solution. It avoids the use of “block estimates” and is self-contained. We apply it to the simple exclusion process, the zero range process, and the Ginzburg-Landau process with Kawasaki dynamics. This is a joint work with Daniel Marahrens and Angeliki Menegaki (IHES).

Modern atmospheric and ocean science require sophisticated geophysical fluid dynamics models. Among them, stochastic partial

differential equations (SPDEs) have become increasingly relevant. The stochasticity in such models can account for the effect

of the unresolved scales (stochastic parametrizations), model uncertainty, unspecified boundary condition, etc. Whilst there is an

extensive SPDE literature, most of it covers models with unrealistic noise terms, making them un-applicable to

geophysical fluid dynamics modelling. There are nevertheless notable exceptions: a number of individual SPDEs with specific forms

and noise structure have been introduced and analysed, each of which with bespoke methodology and painstakingly hard arguments.

In this talk I will present a criterion for the existence of a unique maximal strong solution for nonlinear SPDEs. The work

is inspired by the abstract criterion of Kato and Lai [1984] valid for nonlinear PDEs. The criterion is designed to fit viscous fluid

dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm [2015]. As an immediate application, I show that 

the incompressible SALT 3D Navier-Stokes equation on a bounded domain has a unique maximal solution.

This is joint work with Oana Lang, Daniel Goodair and Romeo Mensah and it is partially supported by European Research Council (ERC)

Synergy project Stochastic Transport in the Upper Ocean Dynamics (https://www.imperial.ac.uk/ocean-dynamics-synergy/

It the talk, various conditions of local regularity of axisymmetric suitable weak solutions, including the so-called slightly supercritical ones, will be discussed.

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni.
The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. 

Most of SDE models in epidemics, ecology, biology, finance etc. are highly nonlinear and do not have explicit solutions. Monte Carlo simulations have played a more and more important role. This talk will point out several well-known numerical schemes may fail to preserve the positivity or moment of the solutions to SDE models. Reliable numerical schemes are therefore required to be designed so that the corresponding Monte Carlo simulations can be trusted. The talk will then concentrate on new numerical schemes for the well-known stochastic Lotka--Volterra model for interacting multi-species. This model has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler-Maruyama (EM) applied to it do not preserve its positivity. The aim of this talk is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM).

Distribution dependent equations (or McKean—Vlasov equations) have found many applications to problems in physics, biology, economics, finance and computer science. Historically, equations with either Brownian noise or zero noise have received the most attention; many well known results can be found in the monographs by A. Sznitman and F. Golse. More recently, attention has been paid to distribution dependent equations driven by random continuous noise, in particular the recent works by M. Coghi, J-D. Deuschel, P. Friz & M. Maurelli, with applications to battery modelling. Furthermore, the phenomenon of regularisation by noise has received new attention following the works of D. Davie and M. Gubinelli & R. Catellier using techniques of averaging along rough trajectories. Building on these ideas I will present recent joint work with L. Galeati and F. Harang concerning well-posedness and stability results for distribution dependent equations driven first by merely continuous noise and secondly driven by fractional Brownian motion.