Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams

TL;DR

This paper presents a new monotone discretization method for the 3D Monge-Ampere operator that combines the simplicity of Wide Stencils with improved accuracy from power diagram techniques, ensuring convergence and efficiency.

Contribution

It introduces a novel discretization scheme that enhances accuracy over traditional methods while maintaining convergence guarantees for 3D Monge-Ampere equations.

Findings

Scheme is consistent when the Hessian condition number is bounded.

Global convergence of the damped Newton solver is established.

Numerical experiments demonstrate the scheme's efficiency in three dimensions.

Abstract

We introduce a monotone (degenerate elliptic) discretization of the Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is consistent provided the solution hessian condition number is uniformly bounded. Our approach enjoys the simplicity of the Wide Stencil method, but significantly improves its accuracy using ideas from discretizations of optimal transport based on power diagrams. We establish the global convergence of a damped Newton solver for the discrete system of equations. Numerical experiments, in three dimensions, illustrate the scheme efficiency.