Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
This work provides a more accurate and practical numerical scheme for solving the Monge-Ampere equation in 3D, which is important for applications in optimal transport, image processing, and geometric optics.
The authors propose a monotone discretization of the 3D Monge-Ampere operator on Cartesian grids that combines the simplicity of Wide Stencil methods with improved accuracy from power diagram-based optimal transport discretizations. They prove global convergence of a damped Newton solver and demonstrate efficiency through 3D numerical experiments.
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.