Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

TL;DR

This paper proves the uniqueness and convergence of a finite difference discretization for the Monge-Ampère equation with prescribed asymptotic cone, using a damped Newton's method and providing numerical validation.

Contribution

It provides the first proof of uniqueness for a discretization of the Monge-Ampère equation with prescribed asymptotic cone and establishes convergence of a damped Newton's method.

Findings

Proved uniqueness of the discretized solution.

Established convergence of the damped Newton's method.

Validated convergence through numerical experiments.

Abstract

For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Amp\`ere equation, we give the first proof of uniqueness. The boundary condition is enforced through the use of the notion of asymptotic cone while the differential operator is discretized based on a discrete analogue of the subdifferential. We establish the convergence of a subsequence of a damped Newton's method for the nonlinear system resulting from the discretization, thereby proving the existence of a solution. Using related arguments we then prove that such a solution is necessarily unique. Convergence of the discretization as well as numerical experiments are given.