Numerical solution of the Monge–Kantorovich problem by density lift-up continuation
Bouharguane, Afaf1 ; Iollo, Angelo1 ; Weynans, Lisl1
1 Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université de Bordeaux and Equipe-projet MEMPHIS, Inria Bordeaux Sud-Ouest, 33405 Talence, France.
ESAIM: Mathematical Modelling and Numerical Analysis , Optimal Transport, Tome 49 (2015) no. 6, pp. 1577-1592.

We present an numerical method to solve the L2 Monge–Kantorovich problem. The method is based on a continuation approach where we iteratively solve the linearized mass conservation equation, progressively decreasing a constant lift-up to map compact support densities in the limit. A Lagrangian as well as an Eulerian integration scheme are proposed. Several examples relative to the transport of two-dimensional densities are investigated, showing that the present methods can significantly reduce the computational effort.

Reçu le :
DOI : 10.1051/m2an/2015024
Classification : 68U01, 65K05
Mots-clés : Optimal transport, Monge–Kantorovich problem, numerical solution, Newton method, continuation approach
Bouharguane, Afaf 1 ; Iollo, Angelo 1 ; Weynans, Lisl 1

1 Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université de Bordeaux and Equipe-projet MEMPHIS, Inria Bordeaux Sud-Ouest, 33405 Talence, France. 
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Bouharguane, Afaf; Iollo, Angelo; Weynans, Lisl. Numerical solution of the Monge–Kantorovich problem by density lift-up continuation. ESAIM: Mathematical Modelling and Numerical Analysis , Optimal Transport, Tome 49 (2015) no. 6, pp. 1577-1592. doi : 10.1051/m2an/2015024. https://www.numdam.org/articles/10.1051/m2an/2015024/

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