Monotone discretization of the Monge–Ampère equation of optimal transport

Abstract

We design a monotone finite difference discretization of the second boundary value problem for the Monge-Ampère equation, whose main application is optimal transport. We prove the existence of solutions to a class of monotone numerical schemes for degenerate elliptic equations whose sets of solutions are stable by addition of a constant, and we show that the scheme that we introduce for the Monge-Ampère equation belongs to this class. We prove the convergence of this scheme, although only in the setting of quadratic optimal transport. The scheme is based on a reformulation of the Monge-Ampère operator as a maximum of semilinear operators. In dimension two, we recommend to use Selling's formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization. We show that this approach yields a closed-form formula for the maximum that appears in the discretized operator, which allows the scheme to be solved particularly efficiently. We present some numerical results that we obtained by applying the scheme to quadratic optimal transport problems as well as to the far field refractor problem in nonimaging optics.

A Convergent Quadrature Based Method For The Monge-Ampère Equation

“…Then u h converges uniformly to u as h → 0. This result does apply to the Monge-Ampère equation (1) with Dirichlet boundary conditions (5) under mild assumptions on the data. However, many other Monge-Ampère equations of interest do not possess the strong comparison principle required by the theorem.…”

“…This allows us to produce new monotone approximation schemes via quadrature. Because these schemes are monotone, they fit within several recently developed numerical convergence frameworks [4,5,16,18,19,27]. Moreover, these new schemes offer significant advantages over existing monotone methods in terms of both accuracy and efficiency.…”

“…An early method [28] used a geometric interpretation of weak solutions to design a convergent, but computationally expensive, method for the 2D Monge-Ampère equation. Recently, convergence frameworks have been established for the Monge-Ampère equation with either Dirichlet boundary conditions [16,18,26,27]: (2) u(x) = g(x), x ∈ ∂Ω or the second type boundary condition arising in optimal transport [4,5,19]:…”

“…Circumventing this challenge requires the use of finite difference stencils that are allowed to grow wider as the grid is refined. Several monotone finite difference schemes are now available for the Monge-Ampère equation [5,15,16,24,27]. Because of the wide-stencil nature of these methods, the methods are computationally expensive and typically have low (sub-linear) accuracy.…”

“…Definition 5 (Comparison Principle). The operator (4) satisfies a strong comparison principle if whenever u is an upper semi-continuous subsolution and v a lower semi-continuous supersolution, then u ≤ v on Ω We remark that the Dirichlet problem for the Monge-Ampère equation ( 1), (5) does satisfy a comparison principle under reasonable assumptions on the data. However, this is no longer true when the right-hand side depends on the solution gradient or when the second type boundary condition (3) is considered [18,19].…”