Abstract. A numerical method for the solution of the elliptic MongeAmpère Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary conditions is combined with a finite difference scheme for the Monge-Ampère equation. Newton's method is implemented leading to a fast solver, comparable to solving the Laplace equation on the same grid several times. Theoretical justification for the method is given by a convergence proof in the companion paper [BFO12]. In this paper, the algorithm is modified to a simpler compact stencil implementation and details of the implementation are given. Solutions are computed with densities supported on non-convex and disconnected domains. Computational examples demonstrate robust performance on singular solutions and fast computational times.

Seismic signals are typically compared using travel time difference or L 2 difference. We propose the Wasserstein metric as an alternative measure of fidelity or misfit in seismology. It exhibits properties from both of the traditional measures mentioned above. The numerical computation is based on the recent development of fast numerical methods for the Monge-Ampère equation and optimal transport. Applications to waveform inversion and registration are discussed and simple numerical examples are presented.

The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution.In this article we build a wide stencil finite difference discretization for the Monge-Ampère equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation.Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration.Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable.

Abstract. Full waveform inversion is a successful procedure for determining properties of the earth from surface measurements in seismology. This inverse problem is solved by a PDE constrained optimization where unknown coefficients in a computed wavefield are adjusted to minimize the mismatch with the measured data. We propose using the Wasserstein metric, which is related to optimal transport, for measuring this mismatch. Several advantageous properties are proved with regards to convexity of the objective function and robustness with respect to noise. The Wasserstein metric is computed by solving a Monge-Ampère equation. We describe an algorithm for computing its Frechet gradient for use in the optimization. Numerical examples are given.

Abstract. The numerical solution of the elliptic Monge-Ampère Partial Differential Equation hasbeen a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155-192; Oliker and Prussner, Numer. Math. 54 (1988) which converge even for singular solutions. However, many of the newly proposed methods lack numerical evidence of convergence on singular solutions, or are known to break down in this case. In this article we present and study the performance of two methods. The first method, which is simply the natural finite difference discretization of the equation, is demonstrated to be the best performing method (in terms of convergence and solution time) currently available for generic (possibly singular) problems, in particular when the right hand side touches zero. The second method, which involves the iterative solution of a Poisson equation involving the Hessian of the solution, is demonstrated to be the best performing (in terms of solution time) when the solution is regular, which occurs when the right hand side is strictly positive.