Richard von Mises Lecture 2022stoA


15:00-16:00 Junior Richard-von-Mises-Lecture by Dr. Thomas Eiter
On time-periodic viscous flow around a moving body
16:00 - 16:30 Coffee Break
16:30 - 17:30 Richard-von-Mises-Lecture by Prof. Dr. Christoph Reisinger
Simulation and control of stochastic mean-field models: from starlings over neurons and traders to supercooling


Simulation and control of stochastic mean-field models: from starlings over neurons and traders to supercooling
Prof. Dr. Christoph Reisinger (Oxford University)

Large interacting random systems are widespread in nature, technology, and society, ranging from large flocks of birds, over networks of communicating neurons, to banks interconnected through mutual lending. Classical work by McKean and Vlasov has paved the way for an elegant description of such systems by stochastic differential equations with coefficients that depend on the law of the process itself.

In this talk, we will review recent theoretical and computational advances in the simulation and control of large interacting particle systems in classical and non-classical settings. We will highlight the numerical challenges presented by the numerical approximation of the resulting McKean–Vlasov equations, as well as the construction of optimal control strategies by gradient iterations of the feedback control map where relevant. Our applications from neuroscience include the FitzHugh-Nagumo model for neurons’ action potentials, where we introduce an adaptive time stepping scheme with guaranteed convergence in the presence of a cubic nonlinearity in the dynamics, and two integrate-and-fire-type models, one characterised explicitly by a discontinuous drift function, and another where a singular interaction may be triggered when the potential reaches a threshold. In the latter case, the main theoretical difficulty is to ensure physical principles are obeyed in the scenario where jumps are triggered. We will discuss the resulting subtle aspects of the simulation of related models from credit risk, where defaults can lead to a wipe-out of significant portions of the banking sector, and analyse optimal bail-out strategies a central bank can adopt to prevent such catastrophic systemic events. We close by revisiting the classical supercooled Stefan problem, which describes the phase transition in a liquid cooled below its freezing point. Exploiting a surprising equivalence with the aforementioned credit model that was discovered recently, we can for the first time present a convergent numerical scheme for this classical nonlinear free boundary problem in the regime where the freezing front exhibits a discontinuity, known in the physics literature as blow-up.

On time-periodic viscous flow around a moving body
Dr. Thomas Eiter (Weierstrass Institute for Applied Analysis and Stochastics)

A classical problem in fluid mechanics is the flow around a moving obstacle; one may think of the flow past an airplane wing or around a rotating propeller, two problems which also attracted the interest of Richard von Mises as one of the great pioneers of aeronautical research. Associated phenomena from everyday life are the occurrence of a wake “behind” the obstacle, which is reflected in the asymptotic behavior of the flow, or the transition from a laminar to an oscillating flow when the translational velocity passes a certain threshold. These observations motivate the study of asymptotic properties of time-periodic fluid flow around a moving body.

This talk focuses on the case when the obstacle rotates with prescribed angular velocity and is immersed into an incompressible viscous fluid. A method to investigate asymptotic behavior is to identify the function spaces in which strong time-periodic solutions can be established. To render the associated resolvent problems uniquely solvable, we introduce a framework of homogeneous Sobolev spaces, which reflects the possibly anisotropic behavior of the flow. For the analysis, the resolvent problem is reduced to an auxiliary problem that is a mixture of a time-periodic problem and a simpler resolvent problem, and which can be studied with methods from harmonic analysis on groups. In the case of a pure rotation without translation, uniform resolvent estimates can be derived, and the existence of time-periodic solutions in a framework of absolutely convergent Fourier series follows without further restrictions. In contrast, if both rotation and translation are present, the uniformity of the resolvent estimates requires additional restrictions, and existence of time-periodic solutions only follows if the two present oscillating processes are compatible: The angular velocity associated with the time periodic flow and the rotational velocity of the body must be rational multiples of each other.

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