Titlebook: Optimal Transport; Old and New Cédric Villani Book 2009 Springer-Verlag Berlin Heidelberg 2009 Monge-Kantorovich problem.Optimal transport.

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河潭

Basic propertiesThe proof relies on basic variational arguments involving the topology of weak convergence (i.e. imposed by bounded continuous test functions).

degradation

Cyclical monotonicity and Kantorovich dualityTo go on, we should become acquainted with two basic concepts in the theory of optimal transport. The first one is a geometric property called cyclical monotonicity; the second one is the Kantorovich dual problem, which is another face of the original Monge—Kantorovich problem. The main result in this chapter is Theorem 5.10.

Observe

The Wasserstein distancesAssume, as before, that you are in charge of the transport of goods between producers and consumers, whose respective spatial distributions are modeled by probability measures.

CLOUT

VEN

Osmosis

Solution of the Monge problem I: global approachIn the present chapter and the next one I shall investigate the solvability of the Monge problem for a Lagrangian cost function. Recall from Theorem 5.30 that it is sufficient to identify conditions under which the initial measure . does not see the set of points where the .-subdifferential of a .-convex function . is multivalued.

失眠症

一窝小鸟

SmoothnessThe smoothness of the optimal transport map may give information about its qualitative behavior, as well as simplify computations. So it is natural to investigate the regularity of this map.

没花的是打扰