Convergent Finite Difference Methods for Singular Solutions of the Elliptic Monge-Ampère Equation
Thursday, March 29, 3:30PM – 5PM
The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory. It has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration as well as in the celebrated Optimal Transportation problem.
Recently several groups of researchers (Dean-Glowinski, Feng-Neilan, Brenner-Neilan) have proposed numerical schemes. These schemes lack a convergence proof in general, and can break down near singular solutions. We build a finite difference equation for the Monge-Ampère equation. We prove solutions of the discrete equation converge to the unique viscosity solution of the PDE. Improvements to accuracy are obtained and a fast Newton solver is implemented, reducing the cost of computation of the equation to the equivalent of a few linear solves.
The most effective notion of weak solutions for fully nonlinear elliptic equations is that of viscosity solutions, developed by Crandall, Ishii, and Lions. Viscosity solutions enjoy strong stability properties, and allow for uniform convergence of approximations, along with pointwise error estimates for approximation schemes.
This theory is used to prove convergence of the finite difference method.
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