Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation

TL;DR

This paper develops a finite difference solver for the elliptic Monge-Ampere equation that converges for singular solutions by combining stable monotone schemes with more accurate discretizations, validated through computational experiments.

Contribution

It introduces a hybrid finite difference method that adaptively switches between stable and accurate schemes to handle singular solutions of the Monge-Ampere equation.

Findings

The solver converges for singular solutions.

Hybrid approach improves accuracy in regular regions.

Computational results confirm efficiency and robustness.

Abstract

The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Ampere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.