Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Amp\`ere equation in dimensions two and higher

TL;DR

This paper develops a convergent finite difference scheme for solving the elliptic Monge-Ampère equation in multiple dimensions, ensuring stability and accuracy for singular solutions using a monotone discretization and Newton's method.

Contribution

It introduces a wide stencil monotone finite difference discretization for the Monge-Ampère equation with proven convergence to viscosity solutions and a systematic approach for Newton's method initialization.

Findings

Scheme converges to viscosity solutions in 2D and 3D

Method handles solutions from smooth to non-differentiable

Demonstrates speed and accuracy on various test cases

Abstract

The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the \MA equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable.