On standard finite difference discretizations of the elliptic Monge-Ampere equation

TL;DR

This paper analyzes the convergence of standard finite difference methods for solving the Monge-Ampere equation, proposing an algorithm that is efficient for smooth and non-smooth solutions, with proven uniform convergence on compact subsets.

Contribution

It introduces a regularization and discretization approach for the Aleksandrov solution and proposes a fast, robust algorithm with convergence guarantees.

Findings

Uniform convergence of discrete solutions to an approximate problem

Algorithm outperforms Newton's method for smooth solutions

High accuracy achieved for non-smooth solutions

Abstract

Given an orthogonal lattice with mesh length h on a bounded convex domain, we propose to approximate the Aleksandrov solution of the Monge-Ampere equation by regularizing the data and discretizing the equation in a subdomain using the standard finite difference method. The Dirichlet data is used to approximate the solution in the remaining part of the domain. We prove the uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain. The result explains the behavior of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions. We propose an algorithm which for smooth solutions appears faster than the popular Newton's method with a high accuracy for non smooth solutions. The convergence of the algorithm is independent of how close to the numerical solution the initial guess is, upon rescaling the equation and given a user's measure of the closeness of an initial guess.