The Monge–Ampère equation and its link to optimal transportation

By Guido De Philippis, Alessio Figalli. 2013

200 citations recorded by the source corpus.

We survey old and new regularity theory for the Monge-Ampere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampere type equations arising in that context.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e601976124a14ffd25aef38b31d5b1f6234a49fd

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Global regularity for the Monge-Ampère equation with natural boundary condition (2018, background): “It is a fundamental problem in the area and received much attention due to its wide range of applications, such as in fluid mechanics, meteorology, image recognition, reflector design, and also in geometry and probability [13, 15, 25, 32, 33].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “For a comprehensive survey of the relations of the Monge-Ampére equation with optimal transport and its regularity properties, see [149].”
On the continuity of center-outward distribution and quantile functions (2018, background): “We note that, although several fundamental results have been obtained in the last 25 years on the regularity of optimal transport maps (see [8, 9] for a survey), the proof of the above-mentioned property is rather delicate, due to the fact that the density of Ud is singular at the origin whenever d ”
Plugin estimation of smooth optimal transport maps (2021, methodology): “Further smoothness properties of Brenier potentials, and therefore of optimal transport maps, have been studied via regularity theory of partial differential equations of the Monge-Ampère type, and we refer to De Philippis and Figalli (2014); Figalli (2017) for surveys.”
Consistent estimation of a population barycenter in the Wasserstein space (2013, unknown): “For a general setting, we refer to the review in [16] for a discussion on sufficient assumptions on the smoothness of Ω, μθ and μ0 which ensure that the optimal mapping Tθ is a C1 diffeomorphism.”
Characterization of barycenters in the Wasserstein space by averaging optimal transport maps (2012, background): “2) holds, we refer to the discussion in [18].”
Quantitative stability of optimal transport maps under variations of the target measure (2021, unknown): “3 in [13]), the optimal transport map Tα,β between ρ and μα,β is the gradient of a strongly convex potential φα,β belonging to C2(X ) and is actually a diffeomorphism between X and Yα.”