Richard von Mises Lecture 2023stoA


14:15-15:15 Junior Richard-von-Mises-Lecture by Sascha Gaudlitz
Statistical theory for stochastic partial differential equations
15:15 - 15:45 Coffee Break
15:45-16:45 Richard-von-Mises-Lecture by Prof. Dr.Jean-Michel Coron
Boundary stabilization of 1-D hyperbolic systems
16:45-17:00 Discussions


Boundary stabilization of 1-D hyperbolic systems
Prof. Dr. Jean-Michel Coron (Université Pierre et Marie Curie )

Hyperbolic systems in one-space dimension appear in various real-life applications (navigable rivers and irrigation channels, heat exchangers, chemical reactors, gas pipes, road traffic, chromatography, ...). This presentation will focus on the stabilization of these systems by means of boundary controls. Stabilizing feedback laws will be constructed. This includes explicit feedback laws that have been implemented for the regulation of the rivers La Sambre and La Meuse. The presentation will also address robustness issues, the case where source terms exist and the case where optimal time stabilisation is considered.

Statistical theory for stochastic partial differential equations
Sascha Gaudlitz

Richard von Mises described mathematics in his article Zur Einführung: Über die Aufgaben und Ziele der angewandten Mathematik (1921) as “a chain of heavily intertwined links”, none of which can be removed completely without disrupting the integrity of the whole. Over a century later, this notion remains profoundly relevant, exemplified by the emerging field of statistical theory for stochastic partial differential equations (SPDEs). Starting from practical questions, such as the optimal calibration of SPDEs to real-world data, emerges the need for sophisticated techniques drawn from diverse disciplines including mathematical statistics, probability and PDE theory. The talk begins with an overview of SPDEs, encompassing their modeling aspects and theoretical properties. Subsequently, we will review the current state of the art on the statistical theory for SPDEs. A rich class of SPDEs are parabolic second order semi-linear equations that are driven by Gaussian noise. We focus on the estimation of the non-linear reaction term, which encapsulates essential features of the underlying dynamics, such as multiple steady points. The calibration procedure presented demonstrates several desirable properties. Firstly, the estimation error can be bounded in terms of the diffusivity of the system and decreases with it. Secondly, the estimation error exhibits asymptotic normality, and our procedure achieves optimality within the class of all possible estimators. In the final part of the talk, we will see that the proof of these results is intricate due to the non-linear behavior of the SPDE and the resulting non-Gaussianity of many involved distributions. To overcome this challenge, we employ the Malliavin calculus, which allows for sharp control of non-linear functionals of Gaussians. The corresponding Poincaré inequality is the key building block for controlling the variance of the estimation error.

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