OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

More documents

Recommendations

Info

To overcome some of these difficulties it isconvenient to study a slightly different problemwhere we have the freedom to distributea given quantity of material without the restrictionof designing a domain Ω. They arecalled Mass Optimization Problems andconsist in finding, given a force field f in R n(which may possibly concentrate on sets oflower dimension), the mass distribution µwhose “resistance” to f is maximal.Constraints:• given mass, ∫ dµ = m• possible design region D given,with spt µ ⊂ D• possible support region Σ given,Dirichlet region.13
Optimization criterium:elastic complianceAs before for every admissible mass distributionµ we consider the energy{ ∫E(µ) = inf j(Du) dµ − 〈f, u〉 :}u smooth, u = 0 on Σand the elastic compliance C(µ) = −E(µ).Then the mass optimization problem is{∫ }min C(µ) : spt µ ⊂ D, dµ = m .Elasticity: u and f vector-valued,j(z) = β|z ∗ | 2 + α 2 |trz∗ | 2 .Conductivity: u and f scalar,j(z) = 1 2 |z|2 .14
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 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 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
show all

OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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

Delete template?

Save as template?