Numerical solution of the second boundary value problem for the Elliptic Monge-Ampère equation

This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampère equation. The boundary conditions correspond to the optimal transportation of measures supported on two domains, where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit and non-local. These boundary conditions are reformulated as a nonlinear Hamilton-Jacobi PDE on the boundary. This formulation allows us to extend the convergent, wide stencil Monge-Ampère solvers proposed by Froese and Oberman to this problem. Several non-trivial computational examples demonstrate that the method is robust and fast. Key-words: Optimal Transportation, Monge-Ampère Equation ∗ INRIA, Domaine de Voluceau Rocquencourt, 78153 Rocquencourt, France † Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada ‡ Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada ha l-0 07 03 67 7, v er si on 1 4 Ju n 20 12 Résolution numérique du deuxième problème aux limites pour l’équation de Monge Ampère Résumé : Cet article présente une méthode de résolution numérique pour une équation de Monge Ampère elliptique non-linéaire. Les conditions aux limites particulières correspondent au problème du transport optimal entre deux mesures dont au moins un des supports est convexe. La difficulté posée tient au caractère implicite et non local de ces conditions aux limites. Nous proposons de les reformuler comme une équation de Hamilton-Jacobi sur le bord. Ceci permet d’étendre les schémas de type ”wide-stencil” et résultats de convergence associés de Froese et Oberman à ce problème. Plusieurs cas tests, certains non triviaux, démontrent la rapidité et robustesse de la méthode. Mots-clés : Transport Optimal, Equation de Monge-Ampère ha l-0 07 03 67 7, v er si on 1 4 Ju n 20 12 NUMERICAL SOLUTION OF THE SECOND BOUNDARY VALUE PROBLEM FOR THE ELLIPTIC MONGE-AMPÈRE EQUATION J.-D. BENAMOU(INRIA-ROCQUENCOURT, FRANCE), B. D. FROESE (SFU, CANADA), AND A. OBERMAN (SFU, CANADA) Abstract. This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampère equation, with boundary conditions corresponding to the optimal transportation of measures supported on two domains, X and Y , where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit. These boundary conditions are reformulated as a nonlinear HamiltonJacobi PDE on the boundary. This formulation allows us to extend the convergent, wide stencil Monge-Ampère solvers proposed in [FO11a] to this problem. Several non-trivial computational examples demonstrate that the method is robust and fast. This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampère equation, with boundary conditions corresponding to the optimal transportation of measures supported on two domains, X and Y , where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit. These boundary conditions are reformulated as a nonlinear HamiltonJacobi PDE on the boundary. This formulation allows us to extend the convergent, wide stencil Monge-Ampère solvers proposed in [FO11a] to this problem. Several non-trivial computational examples demonstrate that the method is robust and fast.