Robust solvers for the Monge-Ampere equation and other fully nonlinear elliptic equations

The Monge-Ampere equation is a classical fully nonlinear elliptic equation. While its origins go back hundreds of years, there has been renewed interest in this equation because of the applications to image warping and optimal transportation.

These applications require robust (convergent) solvers for the equation. However building solvers has been notoriously difficult, with a number of recent contributions by several groups of authors yielding unsatisfactory results.

In this talk, we will begin by giving some background on the equation, and on the different attempts which have been made using ideas from Classical Convex Analysis, Finite Elements, and finally, finite difference methods.

In a first paper, we build convergent finite difference schemes for the equation.

After the renewed interest in the equation, we developed some new solvers, with two goals:

1. building the simplest possible solver to see when it would break down (justifying the use of the convergent solver)
2. building a model of the other methods to see when those break down (showing that the other methods can fail)

• the obstacle problem,
• the infinity Laplacian,
• the equation for motion of level sets by mean curvature,
• the Pucci maximal and minimial equations,
• an equation for the convex envelope.