Speaker: Gerard Awanou, University of Illinois, Chicago

Title: Discrete Aleksandrov solutions of the Monge-Ampere equation

Date: 08/31/2018

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

A discrete analogue of the Dirichlet problem of the Aleksandrov theory of the Monge-Amere equation is derived in this paper. The discrete solution is not required to be convex, but only discrete convex in the sense of Oberman. We prove that the uniform limit on compact subsets of discrete convex functions which are uniformly bounded and which interpolate the Dirichlet boundary data is a continuous convex function which satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. We obtain the first proof of convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition, as proposed by Benamou and Froese, can be used to get a good initial guess for geometric methods solving optimal transport between two measures.