Traffic plans Marc Bernot a , Vicent Caselles b , Jean-Michel Morel c - PowerPoint PPT Presentation

Traffic plans Marc Bernot a , Vicent Caselles b , Jean-Michel Morel c October 21, 2004 a CMLA, ENS Cachan, bernot@cmla.ens-cachan.fr b Universitat Pompeu-Fabra, vicent.caselles@upf.edu c CMLA, ENS Cachan, morel@cmla.ens-cachan.fr 1

Many systems designed by humans can be viewed as supply-demand distribution networks designed to transport goods from one place (the supply) to another (the demand). This is obviously the case with simple good distribution networks such as water distribution, electric power supply, etc. The same can be said of many natural flow networks which connect a finite size volume to a source. This happens for example with irrigation networks, actual plants and trees, bronchial systems or cardiovascular systems. The TRAFFIC PROBLEM has the same characteristics, but it specifies “who goes where". 2

PLAN • Irrigation versus transportation • Three models for irrigation • An extension : the traffic problem • A traffic plan model • Technical issue : parameterization of measures • Existence results, comparison 3

Figure 1: Three rivers 4

Figure 2: Retina 5

Figure 3: Fibers and parallel irrigation 6

Figure 4: Maple leaf, palmately veined 7

Figure 5: Traffic problem : who goes where matters ! 8

The Monge-Kantorovitch problem. • µ + and µ − , measures on R N : the supply and demand mass distributions. • Unknown : transference plan π on R N × R N where π ( A × B ) represents the amount of mass going from A to B • cost function c : R N × R N → R where c ( x, y ) is the cost of transporting a unit mass from x to y . � • question : minimize R N × R N c ( x, y ) dπ ( x, y ) . 9

� � � � ��� ��� � � � ��� ��� � � � � Figure 6: The transport from δ x to 1 2 ( δ y 1 + δ y 2 ) . Monge-Kantorovich versus Q. Xia’s and Maddalena-Solimini’s model. 10

Irrigation versus transportation. Qinglan Xia : In shipping two items from nearby cities to the same far away city, it may be less expensive to first bring them into a common location and put them on a single truck for most of the transport. In this case, a "Y shaped" path is preferable to "V shaped" path". 11

Assume that a tube or a branch e bifurcates into two smaller tubes e 1 et e 2 . Then, by Kirchhoff’s law, the flow w ( e ) = w ( e 1 ) + w ( e 2 ) . If α = 1 , there is no loss of energy in this bifurcation, while if α < 1 , w ( e ) α < w ( e 1 ) α + w ( e 2 ) α . In fluid mechanics : Poiseuille’s law, according to which the resistance of a tube increases when it gets thinner. 12

Qinglan Xia’s formalization • start with finite graphs viewed as one-dimensional flat currents G with non-integer multiplicity satisfying ∂G = µ + − µ − • multiplicity w ( e ) of each edge e : fluid flow • energy E α ( G ) = � e edge of G w ( e ) α length ( e ) , where 0 < α < 1 . • minimizers are flat currents with some regularity. 13

Lagrangian formulation ( Maddalena and Solimini) • single source supply µ + = δ S . • set of particles Ω starting at S • tree (filtration) of their “fibers" or trajectories, χ ( ω, · ) • χ ( ω, t ) is the location of a particle ω ∈ Ω at time t • irrigated measure µ − of a volume is the probability for a fiber to stop in this volume. R + | [ χ ( ω, t )] χ | α − 1 | ˙ � � • energy E ( χ ) = χ ( ω, t ) | dtdω, Ω • Both functionals coincide on trees. Minimizers are the same. 14

� � � � � � � � � � � � � � Figure 7: Maddalena-Morel-Solimini’s versus Qinglan Xia’s model of the irrigation problem with µ + = δ x and µ − = 2 5 δ y 1 + 2 5 δ y 2 + 1 5 δ y 3 . The two geometric objects are the same but on the left-hand side, once fibers separate, they are considered to be separated until they stop. 15

Paths in Wasserstein spaces (Brancolini, Buttazzo, Santambrogio) • irrigate µ − from µ + means defining a path γ ( t ) where γ ( t ) is a probability measure • γ ′ ( t ) is computed in the Wasserstein metric � • the energy is J ( γ ( t )) | γ ′ ( t ) | dt , where J is an energy on probability measures • for instance, J ( � a i δ i ) = � i a α i . 16

• Qinglan Xia : Objects are finite graphs. Energy then extended by relaxation on rectifiable 1-dimensional currents. Problem: this completion has no simple description. • Maddalena-Solimini : Each particle follows a path from the source. A probability space of paths, branching modelled by a filtration. The energy is computed as a function of flow on branches of the tree. • Brancolini, Buttazzo, Santambrogio : irrigate µ − from µ + means defining an optimal path γ ( t ) in a Wasserstein spaces of probabilities. • Objects proposed : probability measure on the set of paths (not a tree). Energy is the same as Maddalena-Solimini. Gain : traffic problem treated in the same formalism as the irrigation problem. 17

B. SAPOVAL : valuable information, documentation and conversations. M. BERNOT, V. CASELLES and J-M. MOREL, Are there infinite irrigation trees? Accepted in Journal of Mathematical Fluid Mechanics. A. BRANCOLINI, G. BUTTAZZO and F. SANTAMBROGIO, Path functionals over Wasserstein spaces , preprint, CVGMT. F. MADDALENA, S. SOLIMINI and J.M. MOREL A variational model of irrigation patterns , Interfaces and Free Boundaries Volume 5, Issue 4, (2003) pp. 391-416. Q. XIA, Optimal paths related to transport problems , Commun. Contemp. Math. 5 , (2003), pp. 251-279. Q. XIA, Interior regularity of optimal transport paths , Calculus of Variations and Partial Differential Equations, 20 , No. 3, (2004), pp. 283-299. 18

��� � � � � � � � � ��� ��� � ��� ��� ��� � � � � � � � � Figure 8: Irrigation versus traffic. Left : classical irrigation. Right : x 1 is specified to go on y 2 and x 2 on y 1 19

The Traffic problem with prescribed transference plan • Let K be a set of paths in X . We define a traffic plan µ as a probability measure on K � • T ( γ ) = stopping time ; K T ( γ ) dµ ( γ ) < ∞ • With any traffic plan µ is associated a transference plan, that is to say a probability measure on X × X that we denote by π µ and define by � < π µ , φ > := φ ( γ (0) , γ ( T ( γ ))) dµ ( γ ) , K where φ ∈ C ( X × X, R ) . • The problem is to find an optimal traffic plan with prescribed π µ . (Same cost functional). 20

� ������ ���� � � ��� � � ��� � � �� � � ��� � � �� ��� ��� ����� � ��� � ��� � � � ���� � � ��� � ������ � ��� � ��� ����� � � Figure 9: Three traffic plans and their associated embedding. 21

A compact set of paths Let X ⊂ R N be a compact set. Let us denote by K the set of 1-Lipschitz maps γ : R + → X . 1 d ( γ, γ ′ ) := sup k || γ − γ ′ || L ∞ ([0 ,k ]) . k ∈ N ∗ Let γ ∈ K . We define its stopping time as T ( γ ) := inf { t : γ constant on [ t, ∞ [ } . The metric space ( K, d ) is compact (Ascoli-Arzela). 22

Traffic plan • A traffic plan µ as a probability measure on K . � • TP C ( X ) is the set of traffic plans µ such that K T ( γ ) dµ ( γ ) ≤ C . We don’t want the average transportation time to be infinite! 23

• With any traffic plan µ is associated a transference plan, that is to say a probability measure on X × X , � < π µ , φ > := φ ( γ (0) , γ ( T ( γ ))) dµ ( γ ) , K where φ ∈ C ( X × X, R ) . • We denote by TP ( π ) the set of traffic plans µ such that π µ = π . This is the set of traffic plans with prescribed transference plan . • Irrigating measure µ − and irrigated measure µ + simply are the marginals of π . • We denote by TP ( ν + , ν − ) the set of traffic plans µ such that µ + = ν + and µ − = ν − . This formalizes the irrigation problem . 24

The traffic problem is the following : given two measures ν + and ν − , and a transference plan π between those measures, we look for minimizers of E with this prescribed transference plan. The irrigation problem is the less constrained case where we specify globally the supply and the demand. 25

Parameterization of a probability measure One can associate with any probability measure a system of "elementary particles" such that µ n ⇀ µ becomes "almost every elementary particle of µ n tends to an elementary particle of µ ". 26

� � � � � � �� � ��� � � � ��� � � �� � � � � � �� � ��� � � � ��� �� � � � � � � �� � ��� � � � ��� � �� � � � �� � ��� � � � ��� ���� � � � Figure 10: Weak convergence of measures becomes almost everywhere convergence of their parameterizations on [0 , 1] . 27

Parameterization of probability measure on [0 , 1] . • µ is a probability measure on K . • λ is the Lebesgue measure on [0 , 1] • We call parameterization of µ by λ , a measurable application χ : ω ∈ [0 , 1] → K such that µ = χ # λ , • that is, µ ( A ) = λ ( χ − 1 ( A )) . 28