A numerical method for the elliptic Monge-Amp\`ere equation with transport boundary conditions

The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\`ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary condition. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Amp\`ere equation with Neumann boundary conditions. To enable mappings between variable densities, we extend an earlier discretization of the equation to allow for right-hand sides that depend on gradients of the solution [Froese and Oberman, SIAM J. Numer. Anal., 49 (2011) 1692--1714]. This discretization provably converges to the viscosity solution. The resulting system is solved efficiently with Newton's method. We provide several challenging computational examples that demonstrate the effectiveness and efficiency ($O(M)-O(M^{1.3})$ time) of the proposed method.