Numerical solution of the second boundary value problem for the Elliptic Monge-Ampère equation
This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampére equation. The boundary conditions correspond to the optimal transportation of measures supported on two domains, where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit and non-local. These boundary conditions are reformulated as a nonlinear Hamilton-Jacobi PDE on the boundary. This formulation allows us to extend the convergent, wide stencil Monge-Ampére solvers proposed by Froese and Oberman to this problem. Several non-trivial computational examples demonstrate that the method is robust and fast.