Mass Transportation on surfaces
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73
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Mass Transportation on surfaces Ludovic Rifford Universite de Nice - Sophia Antipolis Ludovic Rifford Mass Transportation on surfaces
Voir
- lebesgue measure
- optimal transport map
- any measurable map
- monge quadratic
- quadratic cost
Mass
Transportation on surfaces
Ludovic Rifford
Universit´edeNice-SophiaAntipolis
uLdoviciRoffdraMssrTnasportationonsurfaces
measurable
⊂R
,
∀B
onsutiones
Letµ0andµ1beprobability measures with compact supportinRn. We calltransport mapfromµ0toµ1any measurable mapT:Rn→Rnsuch thatT]µ0=µ1, that is µ1(B) =µ0T−1(B),∀Bmeasurable⊂Rn.
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Monge quadratic problem: Study of transport maps T:Rn→Rnwhich minimize thequadratictransport cost Z
se
dµ0(x).
R
|T(x)− |2 x n
usnocafr
Monge quadratic problem of transport maps: Study T:Rn→Rnwhich minimize thequadratictransport cost ZRn|T(x)−x|2dµ0(x).
Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1.
napsroatitnonous
T(x) =rψ(x)µ0a.e. x∈Rn.
Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1. There is a convex functionψ:M→Rsuch that
Monge quadratic problem: Study of transport maps T:Rn→Rnwhich minimize thequadratictransport cost ZR|T(x)−x|2dµ0(x). n
ransassTatioportodiv?yuLroMdRcffiRitaruleg
Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1. There is a convex functionψ:M→Rsuch that
Monge quadratic problem: Study of transport maps T:Rn→Rnwhich minimize thequadratictransport cost ZRn|T(x)−x|2dµ0(x).
Regularity ?
T(x) =rψ(x)µ0a.e. x∈Rn.
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Contre-exemple
trivial
Ludovic
Rifford
Mass
Transp
ortation
on
surfaces
