The second boundary value problem for a discrete Monge-Ampere equation

TL;DR

This paper introduces a discretization method for the second boundary value problem of the Monge-Ampere equation, extending the Oliker-Prussner approach, with proven existence, uniqueness, stability, and convergence of solutions.

Contribution

It proposes a natural discretization of the second boundary condition for the Monge-Ampere equation using a discrete subdifferential, extending existing methods.

Findings

Established existence, uniqueness, and stability of discrete solutions

Proved convergence of the discretization to the continuous problem

Generalized the Oliker-Prussner method for new boundary conditions

Abstract

In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.