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An efficient method for computing solutions to the Optimal Transportation(OT) problem with a wide class of cost functions is presented. The standardlinear programming (LP) discretization of the continuous problem becomesintractible for moderate grid sizes. A grid refinement method results in alinear cost algorithm. Weak convergence of solutions is stablished. Barycentricprojection of transference plans is used to improve the accuracy of solutions.The method is applied to more general problems, including partial optimaltransportation, and barycenter problems. Computational examples validate theaccuracy and efficiency of the method. Optimal maps between nonconvex domains,partial OT free boundaries, and high accuracy barycenters are presented.

An efficient linear programming method for Optimal Transportation