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Section: New Results

Monge-Ampère solver for the Mass Transportation problem and extensions

  • Benamou, Froese (Univ. of Texas at Austin) - We design a scheme for Aleksandrov solution of Optimal Mass Transportation between atomic measure and continuous densities. The idea is to couple the notion of viscosity solution with an adapted sub gradient discretization at dirac points where the notion of Aleksandrov solution is relevant. This would offer a "PDE" alternative to the classical gradient methods based on costly computational geometry tools [61] .

  • Benamou, Collino, Mirebeau (Univ. Paris IX,CNRS) - A new variational formulation of the determinant of a semi-definite positive matrix has been proposed based on the ideas developed in [60] . This leads to a monotone discretisation of the Monge-Ampère operator. A Newton method preserving convexity is currently being tested. The new scheme is more accurate than the wide stencil, currently the state of the art of monotone scheme for the Monge-Ampère equation.

  • Benamou, Froese (Univ. of Texas at Austin), Oberman (Univ. Mc Gill) - When the Optimal Mass Transportation data is not balanced, i.e. the densities do not have equal mass. A natural extension of the optimal transport has been proposed by McCann and Caffareli [30] and revisited by Figalli [41] . It is formulated as an obstacle problem which automatically select the portion of mass corresponding to Optimal Mass Transportation. The numerical resolution of this problem is open and we believe ideas linked the state constraint reformulation contained in paper [6] may be applied to obtain a tractable reformulation.