Solution of Optimal Transportation Problems Using a Multigrid Linear Programming Approach

Abstract

We compute and visualize solutions to the Optimal Transportation (OT) problem for a wide class of cost functions. The standard linear programming (LP) discretization of the continuous problem becomes intractable for moderate grid sizes. A grid refinement method results in a linear cost algorithm. Weak convergence of solutions is established and barycentric projection of transference plans is used to improve the accuracy of solutions. Optimal maps between nonconvex domains, partial OT free boundaries, and high accuracy barycenters are presented.