Density flows and optimal mass transport
Abstract:
We will talk about two closely related topics, Monge-Kantorovich Optimal Mass Transport (OMT) and Schroedinger's Bridges (SB's). These can be viewed as the stochastic control problems to steer the Liouville and Fokker-Planck equations, respectively, between starting and end-point probability distributions, i.e., the problems to regulate the path of uncertain systems between specified marginals. Historically, OMT was introduced by G. Monge in 1781, it was relaxed into a linear programing by L. Kantorovich (1940's), and then recast as a fluid flow problem by Benamou and Brenier (1990's). OMT has since been the cornerstone of advances in physics, econometrics, probability theory, and many other fields. Historically, SB's which amount to a stochastic analogue of OMT, were introduced by E. Schroedinger in 1931 in an attempt to explain Quantum Mechanics in a classical manner and have since proved increasingly pertinent in the same context and for the same applications as OMT. We will explain the connection between the two topics, we will present a numerical scheme for their solution based on a fixed-point iteration (Sinkhorn-like) and the Hilbert metric, and we will highlight the relevance of the two topics in the control of particle ensembles, thermodynamic systems, flows of power spectra, morphing of images. We will also discuss generalizations of OMT and SB's, first to the setting of a discrete space with applications to the transport of resources over networks and, second, to a non-commutative quantum mechanical setting where we will show that the Lindblad equation of open quantum systems is precisely a quantum-density gradient flow of the von Neumann quantum entropy in a suitable non-commutative Wasserstein geometry.