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Abstract: It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the Monge-Ampere equation, a fully nonlinear elliptic PDE. Instead of a traditional boundary condition, the PDE is equipped with a global constraint on the solution gradient, which constrains the transport of mass. Recently, several numerical methods have been proposed for this problem, but no convergence proofs are available. Viscosity solutions have become a powerful tool for analyzing methods for fully nonlinear elliptic equations. However, existing convergence frameworks for viscosity solutions are not valid for this problem. We introduce an alternative PDE that couples the usual Monge-Ampere equation to a Hamilton-Jacobi equation that restricts the transportation of mass. Using this reformulation, we develop a framework for
proving convergence of a large class of approximation schemes for the optimal transport problem. We describe several examples of convergent schemes, as well as possible extensions to more general optimal transportation problems.