Monotone and Consistent discretization of the Monge-Ampere operator
This provides a practical numerical method for solving Monge-Ampere equations, which are important in optimal transport and image processing.
The paper introduces a new discretization of the Monge-Ampere operator that is both monotone and consistent, achieving accuracy and robustness on 2D Cartesian grids. Numerical experiments demonstrate its efficiency.
We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.