OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

More documents

Recommendations

Info

Diffusion case. The following facts in thediffusion case hold:• If µ 0 and µ 1 are in L q (Ω), then they canbe connected by a path γ(t) of finite minimalcost J . The proof uses the displacementconvexity (McCann 1997) which,for a functional F and every µ 0 , µ 1 , is theconvexity of the map t ↦→ F (T (t)), beingT (t) = [(1 − t)Id + tT ] # µ 0 and T an optimaltransportation between µ 0 and µ 1 .• If q < 1 + 1/N then every pair of probabilitiesµ 0 and µ 1 can be connected by apath γ(t) of finite minimal cost J .• The bound above is sharp. Indeed, if q ≥1 + 1/N there are measures that cannotbe connected by a finite cost path (forinstance a Dirac mass and the Lebesguemeasure).59
The previous existence results were basedon two important assumptions:• the compactness of Wasserstein spacesW p (Ω) for Ω compact and 1 ≤ p < +∞;• the estimate like F q ≥ c > 0, that can beobtained when |Ω| < +∞.Both the facts do not hold when Ω is unbounded(indeed, the corresponding Wassersteinspaces are not even locally compact).The same happens when Ω is compact butwe consider the space W ∞ (Ω).Here is a more refined abstract setting whichadapts to the unbounded case.Notice that in the unbounded case theWasserstein spaces W p (Ω) do not contain allthe probabilities on Ω but only those withfinite momentum of order p., that is∫|x| p µ < +∞ .Ω60
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 50 and 51: Now we give a model that takes into
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 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
show all

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?