Numerical solution of the Monge–Kantorovich problem by density lift-up continuation
Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université de Bordeaux
and Equipe-projet MEMPHIS, Inria Bordeaux Sud-Ouest, 33405 Talence, France.
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.
Mathematics Subject Classification: 68U01 / 65K05
Key words: Optimal transport / Monge–Kantorovich problem / numerical solution / Newton method / continuation approach
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