OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Now we give a model that takes into accountwhat occurs in several natural structures(see figures above). Indeed, they presentsome interesting features that should be interpretedin terms of mass transportation.However the usual Monge-Kantorovich theorydoes not provide an explaination to thereasons for which the structures above exist,and it cannot be taken as a mathematicalmodel for them. For instance, if the sourceis a Dirac mass and the target is a segment,as in figure below, the Monge-Kantorovichtheory provides a wrong behaviour.The Monge transport rays.49
Various approaches have been proposed togive more appropriated models:• Q. Xia (Comm. Cont. Math. 2003)through the minimization of a suitablefunctional defined on currents;• V. Caselles, J. M. Morel, S. Solimini, . . .(Preprint 2003 http://www.cmla.enscachan.fr/Cmla/,Interfaces and FreeBoundaries 2003, PNLDE 51 2002) througha kind of analogy of fluid flow in thintubes.• A. Brancolini, G. Buttazzo, E. Oudet, E.Stepanov, (see http://cvgmt.sns.it)through a variational model for irrigationtrees.Here we propose a different approach basedon a definition of path length in a Wassersteinspace.50
Page 1 and 2: OPTIMIZATION PROBLEMS INMASS TRANSP
Page 3 and 4: The Monge problem is thenmin{ ∫ d
Page 5 and 6: Then the class of admissible transp
Page 7 and 8: Monge-Kantorovich problem:{ ∫ }mi
Page 9 and 10: Shape optimization problemsThere is
Page 12 and 13: The shape optimization problem abov
Page 14 and 15: To overcome some of these difficult
Page 16 and 17: In the mass optimization problem th
Page 18 and 19: More precisely, we consider the geo
Page 20 and 21: produces an optimal µ which has a
Page 22 and 23: Variational integrals w.r.t. a meas
Page 24 and 25: Here are some cases where the optim
Page 26 and 27: -2 -1.5 -1 -0.5 0 0.5 1 1.5 21.510.
Page 28 and 29: Problem 1 - Optimal NetworksWe cons
Page 30 and 31: •Large town policy: several ticke
Page 32 and 33: It is interesting to study the opti
Page 34 and 35: The function B can be seen as a con
Page 36 and 37: Here is another case, with a single
Page 38 and 39: • there is a transportation cost
Page 40 and 41: We have then the optimization probl
Page 42 and 43: and so the value of the cost F (a).
Page 44 and 45: is a delicate matter. Indeed, maxim
Page 46 and 47: Some Numerical ComputationsHere are
Page 48 and 49: Optimal sets of length 0.5 and 1 in
Page 52 and 53: We also give a model where the oppo
Page 54 and 55: Theorem Assume:• J is lower semic
Page 56 and 57: ∫∫ ∫J(µ) = f(µ a )dm+ f ∞
Page 58 and 59: Concentration case. The following f
Page 60 and 61: Diffusion case. The following facts
Page 62 and 63: Theorem Let (X, d, d ′ ) be a met
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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