Optimal Transport on Surfaces

Ludovic Rifford

Description

Optimal Transport on Surfaces Ludovic Rifford Universite de Nice - Sophia Antipolis Ludovic Rifford Topics on Optimal Transport (IRMA, 16-17 septembre 2010)

• riemannian surface


• optimal transport


• µ0


• mesurable map


• geodesic distance


• universite de nice


• compact connected

Sujets

• Antipolis
• Transportation theory
• Distance (graph theory)
• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Publié le: 01 septembre 2010
• Nombre de lectures: 22
• Langue: English

Extrait

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Ludovic Rifford

Universite´deNice-SophiaAntipolis

Optimal Transport on Surfaces

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OptimaltiRnonamesnartropldfosanninima eoegthd g bytenoDeddnaenfie q eht rdau sideiscdnctanMeo ∞,b) c y x(y,:) 1 =d 2 aticcostc:M×M→[0 oBowtnevrpnailer 2 ∀y)x, g ( Gi M.y∈x, 1 µnoµs, 0 ad em ,Mnfiilitobabsureymea ] µ g Tµ 10 = isatinfy:TpasM→Marusmelb borelian 1 (B)  ,∀B)Bµ=  0 − Ti(e.µ.( 1 0 (dµ))(x,T(x M c g Z niziminimdna ,)M⊂ malTOpticsonTopioffdrciiRduvo )x L .

a

be

compact

smooth

Riemannian

connected

)

Let (M,g) surface.

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ponsonrtemRinianamnaofinsdlpOitamtlar no,M, 01 µ emus nfiaditymabilresµeasueroBowtnborpnailveGi )B,  B∀oberilnaM⊂.e.µ 1 (B)=µ 0  T −1 ( yfsi T gni = 0 µ ] i( 1 µ blraapemM→T:atMs lTratimaonOppicsdroTiRoffvociL . du x) 0 (dµ))(x,T(x M c Z gniziminimdna ,)

1 2 c(x,y) :=  2 d g (x,y)

Let (M,g) be a smooth compact connected Riemannian surface. Denote byd g the geodesic distance onMand define the quadratic costc:M×M→[0,∞) by

∀x,y∈M.

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Given two Borelian probability measuresµ 0 , µ 1 onM, find a mesurable mapT:M→Msatisfying T ] µ 0 =µ 1 (i.e.µ 1 (B) =µ 0  T −1 (B)  ,∀Bborelian⊂M),

c(x,y 2 1 =:)d g (x,y) 2 ∀x,y∈M.

Let (M,g) be a smooth compact connected Riemannian surface. Denote byd g the geodesic distance onMand define the quadratic costc:M×M→[0,∞) by

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Z

c(x,T(x))dµ 0 (x). M

and minimizing