Project

The international community of Optimal Transportation has rediscovered in recent years a problem posed by E. Schrödinger in 1932, concerning the curious role of probability theory in Quantum Mechanics. A solution to this problem, in terms of stochastic diffusion processes, was given in 1986 by the PI of this project (J-C Zambrini), without ideas or techniques from optimal transport, but the problem and its solution can be - and have been - reinterpreted since then in these terms. This approach (also known today as entropic regularization) has improved considerably , over the last few years, the speed of numerical computations of the solution of Optimal Transport problems in various applied fields ranging from medical imaging to machine learning, fluids models, or mathematical economics. In this way, it has also introduced new and promising connexions between the theory of stochastic processes, Fluids dynamics, Quantum Physics, and Optimal Transport. Our project aims at joining forces of three research groups in Lisbon, Paris, and New York in order to explore further the theoretical, numerical and applied consequences of Schrödinger's problem and its generalizations.

Our research program includes 5 main tasks:

  1. Develop innovative numerical schemes for high dimensional regularized Optimal Transport problems

  2. Elaborate Probabilistic approaches to Fluid Dynamics

  3. Generalize Schrödinger's regularization to mixed quantum states

  4. Introduce notions of integrability for stochastic processes solving Schrödinger's problem

  5. Study and apply the Optimal Transport regularized Inverse Problem in Economics and Imaging

A few keywords:

Optimal transportation; Schrödinger problem; Stochastic optimization; Generalized solutions in fluid dynamics; Stochastic deformation; Entropy;