A variational method for computing numerical solutions of the Monge-Ampere equation
This provides a new numerical approach for solving a challenging nonlinear PDE, but the results are incremental as they confirm convergence on standard tests without demonstrating broad impact or significant performance gains.
The paper introduces a variational method for solving the Monge-Ampere equation by minimizing a convex functional under constraints, proving that the discrete minimizer satisfies the finite difference equation. Numerical tests show convergence to the Aleksandrov solution.
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.