  A numerical method for the solution of the elliptic Monge-Ampere Partial
Differential Equation, with boundary conditions corresponding to the Optimal
Transportation (OT) problem is presented. A local representation of the OT
boundary conditions is combined with a finite difference scheme for the
Monge-Ampere equation. Newton's method is implemented leading to a fast solver,
comparable to solving the Laplace equation on the same grid several times.
Theoretical justification for the method is given by a convergence proof in the
companion paper (Benamou et al., 2012). In this paper, the algorithm is
modified to a simpler compact stencil implementation and details of the
implementation are given. Solutions are computed with densities supported on
non-convex and disconnected domains. Computational examples demonstrate robust
performance on singular solutions and fast computational times.
