Section: Application Domains

Applications of optimal transport

Optimal Transportation in general has many applications. Image processing, biology, fluid mechanics, mathematical physics, game theory, traffic planning, financial mathematics, economics are among the most popular fields of application of the general theory of optimal transport. Many developments have been made in all these fields recently. Two more specific fields:

- In image processing, since a grey-scale image may be viewed as a measure, optimal transportation has been used because it gives a distance between measures corresponding to the optimal cost of moving densities from one to the other, see e.g. the work of J.-M. Morel and co-workers  [66] .

- In representation and approximation of geometric shapes, say by point-cloud sampling, it is also interesting to associate a measure, rather than just a geometric locus, to a distribution of points (this gives a small importance to exceptional “outlier” mistaken points); this was developed in Q. Mérigot’s PhD  [69] in the GEOMETRICA project-team. The relevant distance between measures is again the one coming from optimal transportation. Such approach, combined with evolution of densities mentioned above, may help to perform robust stabilization of set of particles.

McTAO is not directly involved in applications of transport. A starting point may be Alice Erlinger's PhD, in co-supervision with Ludovic Rifford and Robert McCann from the University of Toronto, that will comprehend applications of optimal transportation to the modeling of markets in economy; it is starting in september, 2012.

Applications that would be specific to the type of costs that we consider, i.e. these coming from optimal control are concerned with evolutions of densities under state or velocity constraints. A fluid motion or a crowd movement can be seen as the evolution of a density in a given space. If constraints are given on the directions in which these these densities can evolve, we are in the framework of non-holonomic transport problems. Such approach could be useful to stabilize a large set of particles.