Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher

Abstract

The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that 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. Novel solution methods are required for stability and convergence to the weak (viscosity) solution.In this article we build a wide stencil finite difference discretization for the Monge-Ampère equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation.Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration.Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable.

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

“…Perspectives include the extension of this methodology to other fully nonlinear elliptic equations in two and three dimensions of space in arbitrary domains [7,26].…”

“…In this section, we show that the proposed method based on mixed finite elements applies also to domains with curved boundaries. Note that, in the case of curved boundaries, the use of mixed piecewise linear finite elements provides a substantial simplification compared to using high order finite elements (as in [24,42] for instance), or finite differences [26,27,43].…”

Convergence Rate of a Stable, Monotone and Consistent Scheme for the Monge-Ampère Equation

“…The rate of convergence proven in this paper for a stable, monotone and consistent scheme is a key component of the theory developed in [7] for the convergence of finite difference discretizations to the Aleksandrov solution of the Monge-Ampère equation. It follows from the approach taken therein and the results of this paper, that the discretizations proposed in [1,2] have approximations which converge to the weak solution, as defined in [8], of an approximate problem to Equation (2), even in the general case where Equation (2) does not have a smooth solution. For convergence results in the classical sense, the rate of convergence given here is expected to help establish a rate of convergence for the scheme without a dependence on the smoothness of the solution.…”

“…We obtain a rate of convergence, in the case of smooth convex solutions, of the finite difference schemes introduced in [1,2] for the elliptic Monge-Ampère equation…”

“…The scheme we consider was introduced in [2] but the approach we take also applies to the one introduced in [1]. The consistency error is O(h 2 + dθ) where h is the spatial resolution and dθ the directional resolution.…”