Mass Transportation on surfaces

Ludovic Rifford

Description

Mass Transportation on surfaces Ludovic Rifford Universite de Nice - Sophia Antipolis Ludovic Rifford Mass Transportation on surfaces

• lebesgue measure


• optimal transport map


• any measurable map


• monge quadratic


• quadratic cost

Sujets

• Antipolis
• Lebesgue measure

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Mass

Transportation on surfaces

Ludovic Rifford

Universit´edeNice-SophiaAntipolis

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measurable

⊂R

,

∀B

onsutiones

Letµ 0 andµ 1 beprobability measures with compact supportinR n . We calltransport mapfromµ 0 toµ 1 any measurable mapT:R n →R n such thatT ] µ 0 =µ 1 , that is µ 1 (B) =µ 0  T −1 (B)  ,∀Bmeasurable⊂R n .

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TheBrenieTrehrome letunocyasi 0 losbhritpeesnutiswou (mrBneeihToeer umethatµ r’91) Ass rtna mila mtpapsro tsauexiseoptniquT.erusaeerehtneh e L thtoct emgueseb itno:ψ→MsRcuthahereisaconvexfunc ftsoµmorµot 0 hT . 1 rtfoquheraadccti tionortaanspssTrdraMiRoffvociL ? du tyrilaguRe n .∈R.xe.a 0 µ)x(ψr=)x(T t

Monge quadratic problem: Study of transport maps T:R n →R n which minimize thequadratictransport cost Z

se

dµ 0 (x).

R

|T(x)− | 2 x n

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Monge quadratic problem of transport maps: Study T:R n →R n which minimize thequadratictransport cost Z R n |T(x)−x| 2 dµ 0 (x).

Theorem (Brenier ’91) Assume thatµ 0 is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ 0 toµ 1 .

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TheBrenimrehToeer snonafrusec

T(x) =rψ(x)µ 0 a.e. x∈R n .

Theorem (Brenier ’91) Assume thatµ 0 is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ 0 toµ 1 . There is a convex functionψ:M→Rsuch that

Monge quadratic problem: Study of transport maps T:R n →R n which minimize thequadratictransport cost Z R |T(x)−x| 2 dµ 0 (x). n

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Theorem (Brenier ’91) Assume thatµ 0 is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ 0 toµ 1 . There is a convex functionψ:M→Rsuch that

Monge quadratic problem: Study of transport maps T:R n →R n which minimize thequadratictransport cost Z R n |T(x)−x| 2 dµ 0 (x).

Regularity ?

T(x) =rψ(x)µ 0 a.e. x∈R n .

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Contre-exemple

trivial

Ludovic

Rifford

Mass

Transp

ortation

on

surfaces