Séminaire d'AnalyseOptimal Partial Mass Transportation and Obstacle Monge-Kantorovich Equation
NGUYEN Van Thanh
vendredi 19 mai 2017 14:00 - Tours - Salle 1180 (Bât E2)
Optimal transport consists in finding the optimal way of moving one mass distribution to another in such a way to minimize a certain work. The problem was first proposed by French mathematician G. Monge in 1781 and then L. Kantorovich made fundamental contributions to the problem in the 1940s by relaxing the problem into a linear one. Since the late 80s, this subject has been investigated under various points of view with many surprising applications in partial differential equations (PDEs), differential geometry, image processing, etc. It has been also generalized in various directions. In this work, we are interested in the optimal partial transport which is one of variants of the optimal transport. More precisely, for Finsler distance costs, we introduce equivalent formulations for the characterization of the optimal transportation based on Kantorovich dual problem, minimum flow problem and a PDE of obstacle Monge-Kantorovich type. By studying properties of this PDE and its connection to the optimal partial transport, we prove the uniqueness of the optimal active regions. The equivalent formulations are also used for approximating numerical results via an augmented Lagrangian method. Some numerical simulations are given to validate the approach.