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Variational Heuristics for Optimal Transportation Maps on Compact Manifolds

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Variational Heuristics for Optimal Transportation Maps on Compact Manifolds? Ph. Delanoe† Abstract Variational derivation of the expression of the solution of Monge's problem posed on compact manifolds (possibly with boundary), assuming all data are smooth, the solution is a di?eomorphism and the cost function satisfies a generating type condition. 1 Introduction Let X be a compact connected manifold (all objects are C∞ unless otherwise stated). We assume that, either X has no boundary, or it is a domain contained in some larger manifold. Let Y be a manifold di?eomorphic to X and µ (resp. ?) an everywhere positive probability measure on X (resp. Y ). We say that a map ? : X ? Y pushes µ to ?, and write ?_µ = ?, if the following equality1: ∫ Y h d? = ∫ X (h ? ?) dµ(1) holds for each function h : Y ? R. Furthermore, let c : ? ? X ? Y ? R be a function defined in a domain2 ? projecting onto X (resp. Y ) by the canonical first (resp. second) projection pX (resp. pY ). In this setting, Monge's problem consists in minimizing the functional C(?) = ∫ X c(x, ?(x)) dµ(2) among the Borel maps ? : X ? Y which satisfy the constraint: ?_µ = ? .

  • must specify

  • lebesgue measure

  • divergence operator

  • py ?

  • frechet space

  • cost function

  • di?µ

  • ?v ?

  • cost functional


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Variational Heuristics for Optimal Transportation Maps on Compact Manifolds
1
Ph.Delano¨e
Abstract Variational derivation of the expression of the solution of Monge’s problem posed on compact manifolds (possibly with boundary), assuming all data are smooth, the solution is a diffeomorphism and the cost function satisfies a generating type condition.
Introduction
LetXbe a compact connected manifold (all objects areCunless otherwise stated). We assume that, eitherXhas no boundary, or it is a domain contained in some larger manifold. LetYbe a manifold diffeomorphic toXandµ(resp. ν) an everywhere positive probability measure onX(resp.Ysay that a). We 1 mapϕ:XYpushesµtoν, and writeϕ#µ=ν, if the following equality :   (1)h dν= (hϕ)Y X holds for each functionh:YRlet. Furthermore, c: ΩX×YRbe a 2 function defined in a domain Ω projecting ontoX(resp.Y) by the canonical first (resp. second) projectionpX(resp.pYthis setting, Monge’s problem). In consists in minimizing the functional (2)C(ϕ) =c(x, ϕ(x))X among the Borel mapsϕ:XYwhich satisfy the constraint:
(3)
ϕ#µ=ν .
Optimal transportation is the theory designed for tackling such problems (see [Vil09] and references therein). In general, even though the present data are smooth, one can only solve a relaxed version of Monge’s problem due to Kan-torovich; but if the cost functioncsatisfies a weak form of the generating con-dition stated below, Monge’s problem becomes solvable [Vil09, Theorem 5.30]
2000 MSC: 58E99; 58D05; 58A10;Key-words: Monge’s problem, generating cost func-tion,c-exponential, Euler equation, Helmholtz lemma Supported by the CNRS 1 here, we may allowϕandhto be merely Borel 2 we make no smoothness assumption on its boundary