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

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Traffic plans

Marc Bernot a, Vicent Caselles b , Jean-Michel Morel c October 21, 2004

aCMLA, ENS Cachan, bernot@cmla.ens-cachan.fr bUniversitat Pompeu-Fabra, vicent.caselles@upf.edu cCMLA, ENS Cachan, morel@cmla.ens-cachan.fr

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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".

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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

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Figure 1: Three rivers

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Figure 2: Retina

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Figure 3: Fibers and parallel irrigation

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Figure 4: Maple leaf, palmately veined

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Figure 5: Traffic problem : who goes where matters !

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The Monge-Kantorovitch problem.

µ+ and µ−, measures on RN : the supply and demand mass

distributions.

Unknown : transference plan π on RN × RN where π(A × B)

represents the amount of mass going from A to B

cost function c : RN × RN → R where c(x, y) is the cost of

transporting a unit mass from x to y.

question : minimize
RN×RN c(x, y)dπ(x, y).

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Figure 6: The transport from δx to 1

2(δy1 + δy2). Monge-Kantorovich

versus Q. Xia’s and Maddalena-Solimini’s model.

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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".

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Assume that a tube or a branch e bifurcates into two smaller tubes e1 et

e2. Then, by Kirchhoff’s law, the flow w(e) = w(e1) + w(e2). If α = 1,

there is no loss of energy in this bifurcation, while if α < 1, w(e)α < w(e1)α + w(e2)α. In fluid mechanics : Poiseuille’s law, according to which the resistance of a tube increases when it gets thinner.

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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 Eα(G) =

e edge of G w(e)αlength(e), where 0 < α < 1.

minimizers are flat currents with some regularity.

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Lagrangian formulation ( Maddalena and Solimini)

single source supply µ+ = δS.
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.

energy E(χ) =
Ω
R+ |[χ(ω, t)]χ|α−1| ˙

χ(ω, t)|dtdω,

Both functionals coincide on trees. Minimizers are the same.

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Figure 7: Maddalena-Morel-Solimini’s versus Qinglan Xia’s model of the

irrigation problem with µ+ = δx and µ− = 2

5δy1 + 2 5δy2 + 1 5δy3. 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.

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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( aiδi) =

i aα i . 16

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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.

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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
f 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.

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Figure 8: Irrigation versus traffic. Left : classical irrigation. Right : x1 is

specified to go on y2 and x2 on y1

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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 πµ and define by < πµ, φ >:=

K

φ(γ(0), γ(T(γ)))dµ(γ), where φ ∈ C(X × X, R).

The problem is to find an optimal traffic plan with prescribed πµ.

(Same cost functional).

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Figure 9: Three traffic plans and their associated embedding.

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A compact set of paths Let X ⊂ RN be a compact set. Let us denote by K the set of 1-Lipschitz maps γ : R+ → X. d(γ, γ′) := sup

k∈N∗

1 k ||γ − γ′||L∞([0,k]). 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).

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Traffic plan

A traffic plan µ as a probability measure on K.
TPC(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!

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With any traffic plan µ is associated a transference plan, that is to say

a probability measure on X × X, < πµ, φ >:=

K

φ(γ(0), γ(T(γ)))dµ(γ), where φ ∈ C(X × X, R).

We denote by TP(π) the set of traffic plans µ such that πµ = 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.

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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.

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Parameterization of a probability measure One can associate with any probability measure a system of "elementary particles" such that µn ⇀ µ becomes "almost every elementary particle

f µn tends to an elementary particle of µ".

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Figure 10: Weak convergence of measures becomes almost everywhere

convergence of their parameterizations on [0, 1].

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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)).

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Example

the Dirac mass at 0 is parameterized by the null constant application
n [0, 1].
In the same way, an atomic measure n

1 aiδxi is parameterized by

the piecewise constant function χ(ω) = x1 on [0, a1], χ(ω) = x2 on ]a1, a2] and so on.

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Let (µn)n and µ be probability measures on a compact metric space (K, d). We say that µn tends to µ "pointwise" whenever there exist parameterizations χn and χ of µn and of µ, respectively, such that d(χn(ω), χ(ω)) → 0 almost everywhere in [0, 1]. Theorem 0.1 Let (µn)n be a sequence of probability measures on (K, d). The sequence µn weakly-* converges to µ if and only if µn to µ tends to µ "pointwise".

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Advantages of the probability measure formalization :

compactness of the set of traffic plans is given by the compactness of

the probability measures for weak convergence

this weak convergence yields a pointwise convergence for traffic

plans, which means a convergence path by path

Immediately : If µn ⇀ µ, then πµn ⇀ πµ
If a sequence of traffic plans µn converges to µ , then
K

L(γ)dµ(γ) ≤ lim inf

K

L(γ)dµn(γ).

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Multiplicity of a traffic plan Lower semicontinuity of the cost functional boils down to the upper semicontinuity of the multiplicity: Definition Let µ be a traffic plan. We call multiplicity of µ at a point x ∈ RN the number |x|µ := µ({γ : ∃t, γ(t) = x}). If χ is a parameterization of µ, then |[x]µ| = [x]χ := {ω : ∃t, χ(ω, t) = x}.

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Three semicontinuity properties of the multiplicity Let µn be a sequence of traffic plans with parameterizations χn. If

Ω T(χn(ω))dω ≤ C and χn converges to χ pointwise (or equivalently

µn tends weakly to µ. Then (Maddalena-Solimini)

lim supn |x|µn ≤ |x|µ.
for almost all ω, lim sup |[χn(ω, t)]χn| ≤ |[χ(ω, t)]χ|.
the function x → |x|µ is upper semicontinuous

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Energy of a traffic plan and existence of a minimizer Definition Let α ∈ [0, 1]. We call energy of a traffic plan the functional E(µ) =

Ω
R+ |[χ(ω, t)]χ|α−1| ˙

χ(ω, t)|dtdω, (1) where χ is a parameterization of µ. It’s a generalization of the one used in Maddalena-Solimini. It can be proved to be equivalent to Qinglan Xia’s.

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Existence of a minimizer Proposition If (µn)n is a sequence of traffic plans (whose paths are parameterized by length) in TPC, and µ is a traffic plan such that µn ⇀ µ, then E(µ) ≤ lim inf E(µn). Proposition The problem of minimizing E(µ) in TP(π) admits a solution.

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Properties of the geometric support Let µ be a traffic plan with finite energy :

The range of µ is covered with a countable set of paths. This permits

us to compare our energy with the formulation given by Q. Xia

There exists a loop-free traffic plan ˜

µ with smaller range and same transference plan : π˜

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A change of variable formula Proposition Let χ be a parameterization of a traffic plan µ. Then, E(µ) =

Ω

∞ |[χ(ω, t)]|α−1| ˙ χ(ω, t)| dt dω ≥

RN |[x]χ|α dH1(x).

(2) If we assume, in addition, that χ is loop free, we have E(µ) =

Ω

∞ |[χ(ω, t)]|α−1| ˙ χ(ω, t)| dt dω =

RN |[x]χ|α dH1(x).

(3)

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