A numerical method for the elliptic Monge-Ampère 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-Ampère 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 conditions. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Ampère equation with Neumann boundary conditions. We present a new discretization for the equation, which 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(M1.3) time) of the proposed method.