06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Michela Ottobre (Heriot-Watt University) and Carsten Hartmann (BTU Cottbus-Senftenberg)
2pm-2:45pm Michela Ottobre (Heriot-Watt University): Non-mean-field interacting particle systems
3pm-3:45 pm Carsten Hartmann (BTU Cottbus-Senftenberg): Optimal control of the underdamped Langevin sampler
Michela Ottobre (Heriot-Watt University): Non-mean-field interacting particle systems
The study of interacting particle systems has a very long history and many interesting systems in physics or in the applied sciences are constituted by a large number of particles or agents, (e.g. individuals, animals, cells, robots) that interact with each other. In statistical physics one usually considers simplified models, typically PDE-based models, that retain all the relevant characteristics of the original particle system. Such models were proposed with the intent of efficiently directing human traffic, to optimize evacuation times, to study rating systems for online games, for opinion formation or in control engineering. In this seminar we will consider a population of N particles interacting with each other through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e. with O(N) particles, we consider the a priori more involved case of a sparse network; that is, particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics, with the time-scale of the network described by a parameter ε >0. We show how to combine the averaging (ε→0) and the many particles (N→∞).
Carsten Hartmann (BTU Cottbus-Senftenberg): Optimal control of the underdamped Langevin sampler
The underdamped Langevin equation is a popular computational model in various fields of science (e.g. molecular dynamics, meteorology, or machine learning) that is used to sample from complicated multimodal probability distributions by a combination of (dissipative) Hamiltonian dynamics and diffusion. I will explain how control theory can help to speed up the convergence of an underdamped Langevin equation to its equilibrium probability distribution, specifically in situations in which the diffusion coefficient is small (i.e. low temperature) and the asymptotic properties of the dynamics are dominated by metastability and poor convergence. I will discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower ("target") or the higher ("simulation") temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow.
This talk is based joint work with Tobias Breiten, Lara Neureither and Upanshu Sharma.
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