Optimal transport maps in Monge-Kantorovich problem

By Luigi Ambrosio. 2003

49 citations recorded by the source corpus.

In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence of optimal transport maps. We show that in several cases optimal transport maps can be obtained by a singular perturbation technique based on the theory of $\Gamma$-convergence, which yields as a byproduct existence and stability results for classical Monge solutions.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/69f2ede23b4d850aadb307f2ffe02e5890fa4252

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Riemannian Score-Based Generative Modeling (2022, background): “This approach is motivated by the fact that the optimal transport map takes such an expression (Ambrosio, 2003).”
Regularity of Potential Functions of the Optimal Transportation Problem (2005, background): “The original cost function of Monge was proportional to the distance moved and surprisingly this problem was only recently completely solved [9,29], see also [2].”