The second boundary value problem for a discrete Monge-Ampere equation

TL;DR

This work develops a discretization for the second boundary value problem of the Monge–Ampère equation by leveraging a discrete subdifferential and an asymptotic-cone based convex extension. The scheme uses a Cartesian grid and a stencil-based discrete Monge–Ampère operator, with a discrete extension that implicitly enforces the second boundary condition and mass conservation. The authors prove existence, uniqueness (up to constants in certain cases), and stability of discrete solutions, and establish convergence to the continuous Aleksandrov or viscosity solutions under broad assumptions, with separate treatment for polygonal approximations of $\Omega^*$ and the degenerate case $f\ge 0$. Numerical experiments corroborate the theoretical results and demonstrate robustness and efficiency, including linear complexity when using a fixed stencil and avoidance of power diagrams in 3D. Altogether, the paper provides a rigorous, scalable framework for discretizing and solving the second boundary Monge–Ampère problem in geometric optics and optimal transport.

Abstract

In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

Figures (5)

• Figure 1: A polyhedral angle in $\mathbb{R}^3$. The dashed polygon is a virtual cut of the unbounded set. To emphasize that a polyhedral angle has non zero Lebesgue measure, a filled version is shown.
• Figure 2: Epigraph of $k_{(p,\mu)}$ with a parallel translation as epigraph of $k_{(q,\gamma)}$. The dotted lines at the top of the figure represent virtual cuts of the unbounded epigraphs. As $\gamma < \mu$ in the figure, more of the unbounded epigraph is shown.
• Figure 3: Let $S$ denote the polygon with vertices $A(-1.5,1), B(-1,0), C(1,0)$ and $D(1.5,1)$. The polygon $S$ is the convex hull of its vertices. The polyhedral angle $K_{}$ associated to $\overline {\Omega^*}=[-3,3]$ is the intersection of the half-spaces $\{ \, (x_1,x_2) \in \mathbb{R}^2: x_2 \geq 3 x_1 \, \}$ and $\{ \, (x_1,x_2) \in \mathbb{R}^2: x_2 \geq -3 x_1 \, \}$. Parallel translates $E+K$, $A+K$ and $D+K$ are shown. Put $M=\operatorname{Conv}(S \cup (E+K))$. To visualize $M$, note that $S \subset M$ and $E+K \subset M$. Then draw line segments connecting $A$ or $D$ to points on the boundary of $E+K$. Note that $\overline {M}$ is obtained by sweeping $K$ over $S$. The projection of the convex set $S$ on $\mathbb{R}$ is $[-1.5,1.5]$. The convex set $S$ defines a piecewise linear convex function $u$ on $[-1.5,1.5]$, with graph the lower part of the boundary of $S$. By Lemma \ref{['conv-hull-union']}, $\overline {M}=S+K$ is the convex hull of $S$ and $A+K$. By Theorem \ref{['formula2']}, the piecewise linear convex function on the real line defined by $\overline {M}$, i.e. the convex function with graph the lower part of the boundary of $\overline {M}$, is a convex extension of $u$ and is obtained by the extension formula. By Theorem \ref{['as-poly']}$\overline {M}$ has asymptotic cone $A+K_{}$. The ray with vertex $A$ and slope -3 and the ray with vertex $D$ and slope 3 are called extreme rays. The set $M$ is the convex hull of its vertices $A, B, C$ and $D$ and its extreme rays. Image reproduced from Awanou-ams.
• Figure 4: Constant density on a square mapped to constant density on the unit disc $h=1/2^7$.
• Figure 5: Constant density on a square mapped to the Gaussian $e^{-0.5(x^2+y^2)}$ on the unit disc $h=1/2^8$.

Theorems & Definitions (106)

• definition 1
• definition 2
• definition 3
• theorem 1
• theorem 2
• remark 1
• lemma 1
• proof
• remark 2
• lemma 2