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

TL;DR

This paper introduces a variational numerical method for solving the Monge-Ampere equation, proving convergence of the discrete minimizer to the Aleksandrov solution using monotone schemes.

Contribution

It develops a novel variational approach based on convex functional minimization, ensuring convergence and correctness of solutions for the Monge-Ampere equation.

Findings

Numerical solutions converge to the Aleksandrov solution.

The method works with both standard and monotone discretizations.

Convergence is proven for the discrete minimizers.

Abstract

We present a numerical method for solving the Monge-Ampere 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.