NCSU Differential Equations/Nonlinear Analysis Seminars

Location and Time: SAS 4201, Wednesday 15:00 -16:00

Organizers: Lorena Bociu, Patrick Combettes, Ryan Murray, and Khai T. Nguyen

NEXT TALK

Wednesday, Oct 08, 15:00-16:00, Zoom: Link

Speaker: Sangmin Park, California Institute of Technology
Title: Dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation
Abstract: The Vlasov-Fokker-Planck equation (VFP) describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. Moreover, the geometric structure has possible connections to the conjectured optimal convergence rates of underdamped Langevin Monte Carlo (ULMC), a sampling algorithm known to empirically outperform the (standard) Langevin Monte Carlo. 


This talk will focus on a time-discrete variational scheme for VFP which we introduce to understand the geometric structure of the equation more rigorously. We will begin by introducing the optimal transport problem and the Wasserstein distance, as well as the techniques of gradient flows which form the basis of our variational scheme. After highlighting the connections to ULMC, we will discuss how the proposed variational scheme is (i) consistent with the dissipative Hamiltonian structure, and (ii) (geodesically)-convex at each iteration.

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