Generalised Optimal Transport and Applications – GOTA

This project deals with some generalizations of Optimal Transport problems and their applications. We plan to tackle three main topics: multi-marginal optimal transport (and applications in risk measures and quantum chemistry), multi-population models in urban planning and entropic optimal transport. First, we will investigate connections between risk estimation (understanding the worst case combination of events) and multi-marginal optimal transportation problems. Since these problems are affected by the so-called curse of dimensionality, meaning that they are not numerically tractable, a crucial objective is to characterize the dimension of the solution and to implement new efficient numerical methods.
Second, we propose to develop new tools based on Optimal Transport to deal with theoretical and numerical solution of systems of PDE, involving several populations, which naturally arise, for example, in dynamic of urban system. In the same flavour we also plan to tackle the problem of building urban networks (such as the transportation network) in order to minimise the commuting time of several populations (practitioners, blue collar etc). Finally, we want to focus on the entropic optimal transport problem which has recently become very popular as it is an efficient way to approximate optimal transport problems, especially from a computational viewpoint. However there is the need to better understand the rates of convergence and asymptotic developments of the optimal costs, plans or potentials as the noise parameter vanishes; for the classical optimal transport problem as well as for other regularized optimal transport problems as multi-marginal ones.