A variational method for computing numerical solutions of the Monge-Ampere equation

Abstract

We present a numerical method for solving the Monge-Ampère equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear constraints. When the equation is discretized with a certain monotone scheme, we prove that the unique minimizer of the discrete problem solves the finite difference equation. For the numerical results we use both the standard finite difference discretization and the monotone scheme. Results with standard tests confirm that the numerical approximations converge to the Aleksandrov solution.