[Paper Review] Monotone and Consistent discretization of the Monge-Ampere operator

Jean‐David Benamou, Francis Collino+1 more|arXiv (Cornell University)|Sep 23, 2014

Geometry and complex manifolds6 references4 citations

TL;DR

This paper introduces a novel monotone and consistent finite difference scheme for the Monge-Ampere operator on 2D Cartesian grids by leveraging lattice basis reduction and the Stern-Brocot tree. The method ensures discrete degenerate ellipticity and automatic stencil adaptation, achieving high accuracy and robustness without tuning parameters, even for non-smooth solutions.

ABSTRACT

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.

Motivation & Objective

• Address the lack of consistent and degenerate elliptic schemes for the Monge-Ampere PDE in numerical methods.
• Overcome limitations of existing schemes that are either non-local, only approximately consistent, or require smooth solutions and careful initialization.
• Develop a parameter-free, solution-adapted discretization stencil that ensures both consistency and degenerate ellipticity.
• Utilize advanced arithmetic and geometric tools—lattice basis reduction and the Stern-Brocot tree—to construct a robust numerical scheme on structured grids.

Proposed method

• Construct a finite difference discretization of the Hessian using second-order differences on a 2D Cartesian grid, with boundary adjustments via weighted averages when stencils extend beyond the domain.
• Apply lattice basis reduction to identify optimal, anisotropy-aware stencils that preserve degenerate ellipticity at the discrete level.
• Use the Stern-Brocot tree to generate stencils automatically and adaptively, avoiding arbitrary choices and reducing consistency errors.
• Ensure the scheme is monotone by enforcing that the discrete operator satisfies a comparison principle through geometric constraints on the lattice.
• Maintain consistency under the assumption that the Hessian condition number of the solution is uniformly bounded.
• Integrate the scheme into a viscosity solution framework, enabling convergence guarantees and stable iterative solvers.

Experimental results

Research questions

• RQ1 Can a finite difference scheme for the Monge-Ampere operator be both consistent and degenerate elliptic on a Cartesian grid?
• RQ2 How can the stencil selection process be automated and adapted to the solution’s local anisotropy without introducing tuning parameters?
• RQ3 Can lattice-theoretic tools like the Stern-Brocot tree and lattice basis reduction be effectively applied to improve numerical discretization of nonlinear PDEs?
• RQ4 Does the proposed scheme maintain robustness and accuracy even when the solution lacks high regularity?
• RQ5 What is the impact of geometric and arithmetic structures on the convergence and stability of discrete solutions to the Monge-Ampere PDE?

Key findings

• The proposed MA-LBR scheme is both consistent and degenerate elliptic, ensuring convergence of discrete solutions to the viscosity solution and stability of iterative solvers.
• The scheme achieves automatic, parameter-free stencil generation via the Stern-Brocot tree, eliminating the need for heuristic or user-defined stencil selection.
• Lattice basis reduction enables the construction of anisotropic stencils that respect the invariance of the Monge-Ampere operator under unimodular transformations.
• Numerical experiments confirm the method’s high accuracy and robustness, particularly in cases where traditional schemes fail due to low regularity or poor initialization.
• Theoretical analysis relies on novel tools from discrete geometry, including the geometry of 2D lattices and the arithmetic structure of the Stern-Brocot tree, which are rarely used in numerical analysis.
• The method guarantees a comparison principle at the discrete level, which is essential for convergence in the viscosity solution framework.