Monotone and consistent discretization of the Monge-Ampère operator

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.

Discretization of the 3d monge−ampere operator, between wide stencils and power diagrams

“…Filtered schemes [FO13] combine a monotone and a consistent scheme, and attempt to cumulate their robustness and accuracy; improving either of the constituting schemes will benefit to the filtered combination. A monotone and consistent scheme is introduced in [BCM14], but it is limited to two dimensions. Geometric approaches [OP89] are discussed in the third paragraph.…”

“…for which the solution hessian is almost degenerate), but at the cost of an increased computation time. The choice of stencil is left to the user; let us mention that in the special case of [BCM14] an automatic (solution adaptive, local, anisotropic, and parameter free) stencil construction could be designed. Our numerical experiments show that small stencils, of radius √ 3 or √ 6, see the table page 13, already yield convincing results.…”

Numerical resolution of Euler equations through semi-discrete optimal transport

“…First, the constraint of convexity can be discretized in various ways, none of which is particularly simple or canonical [2,34,35]. For the problem of interest, convexity can also be imposed through the discretization of the Monge-Ampere operator [26,8]. Second, discretizations of the boundary condition ∇u(X) = Y have only been proposed recently [10].…”

“…However, it is desirable to preserve its monotony at the discrete level, in the spirit of [28], which requires more complex implementations [26,8].…”

An augmented Lagrangian approach to Wasserstein gradient flows and applications

“…This method is provably convergent and enables one to use of a Newton method. Note that using monotone finite difference Monge-Ampère solvers as introduced in [10], [9] could be an option but it does not seem to have been tried.…”