Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation

Authors:
Jean-David Benamou;Brittany D. Froese;Adam M. Oberman
Affiliations:
INRIA, Domaine de Voluceau, B.P. 105, 78153 Rocquencourt, France;Department of Mathematics, Simon Fraser University, 8888 University Drive, BC V5A 1S6 Burnaby, Canada;Department of Mathematics, Simon Fraser University, 8888 University Drive, BC V5A 1S6 Burnaby, Canada
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

A numerical method for the solution of the elliptic Monge-Ampere 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-Ampere 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 [4]. Solutions are computed with densities supported on non-convex and disconnected domains. Computational examples demonstrate robust performance on singular solutions and fast computational times.