[PPT] - Optimal regions for congested transport Giuseppe Buttazzo PowerPoint Presentation

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Optimal regions for congested transport

Giuseppe Buttazzo Dipartimento di Matematica Universit` a di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it

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Joint work with: Guillaume Carlier - Paris Dauphine Serena Guarino - University of Pisa

(to appear) available at: http://cvgmt.sns.it arxiv.org

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We consider a geographical region Ω in which two densities f+ and f− are given; for in- stance we may think f+ is the distribution

working places. We assume

We also assume that the transport in Ω is congested and that the congestion function is given by a convex nonnegative superlinear function H.

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It is known that in this case the traffic flux σ, in the stationary regime, reduces to a min- imization problem of the form (Beckmann model) min

Γf =

being f = f+ − f−. See for instance Brasco- Carlier, Carlier-Jimenez-Santambrogio for de- tails on the model.

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The pure Monge’s problem corresponds to H(s) = |s| in which no congestion occurs, and the congestion effect is higher for larger functions H. In some cases, the transportation cost can be +∞ if the source and target measures f+ and f− are singular; for instance, if H has a quadratic growth, in order to have a finite cost it is necessary that the signed measure f = f+ − f− be in the dual Sobolev space H−1.

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Our goal is to reduce the congestion acting

tions H1 and H2 are given, with H1 ≤ H2, and the goal is to find an optimal region C ⊂ Ω where we enforce a traffic conges- tion reduction. Since reducing the congestion in C is costly, a penalization term m(C) is added, to de- scribe the cost of the improvement, then pe- nalizing too large low-congestion regions.

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For every region C we then consider the shape function F(C) = min

region amounts to the minimization problem min

Much of the analysis below depends on the penalization function m(C).

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The case m(C) = k Per(C) Theorem For every k > 0 there exists an

Indeed, a minimizing sequence Cn has a uni- formly bounded perimeter and so we may as- sume that Cn tends strongly in L1 to some set C. The perimeter is L1-lower semicon- tinuous; moreover, the optimal σn ∈ Γf pro- viding the value F(Cn) =

H2(σn) dx +

H1(σn) dx

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are weakly L1 compact, due to the super- linearity of the congestion functions (de La Vall´ ee Poussin theorem). The conclusion now follows from the strong-weak lower semi- continuity theorem for integral functionals. The necessary conditions of optimality are: σ =

∇H∗

in C ∇H∗

in Ω \ C where uint and uext solve the PDEs

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− div

∇H∗

− div

∇H∗

with the transmission condition across ∂C ∇H∗

Performing the shape derivative on ∂C we also obtain on ∂C ∩ Ω

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H2

≤ H2

where HC is the mean curvature on ∂C. Since H1 ≤ H2 this gives that HC ≥ 0. If d = 2 and Ω is convex, replacing C by its convex hull diminishes the perimeter and also the congestion cost, so the optimal C are convex.

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The case m(C) = k|C| Passing from sets C to density functions θ(x) ∈ [0, 1] we obtain the relaxed formulation min

The minimization with respect to θ is straight- forward and the optimal θ is θ = 1H1(σ)+k<H2(σ), which reduces the problem to min

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Since the function H = H2 ∧ (H1 + k) is not convex, a further relaxation gives finally the problem min

If σ is a solution, we have

improvement for low congestion is needed;

is in this region we have to spend a lot to improve the congestion;

sources proportionally to θ(x) ∈]0, 1[.

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If H1 and H2 only depend on |σ| we get θ(x) =

if |σ| ≤ r1

if r1 < |σ| < r2 1 if |σ| ≥ r2 where r1 and r2 can be explicitly computed from H1 and H2: r1 = max. solution of H∗∗(r) = H2(r) r2 = min. solution of H∗∗(r) = H1(r) + k. Some numerical computations can be made when H1 and H2 are quadratic.

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The problem in this case is similar to the two-phase shape optimization problem, for which we refer to the book by Allaire [Springer 2001]. We take: H1(σ) = a|σ|2, H2(σ) = b|σ|2 with a < b. Then we have H∗(ξ) = ξ2 4b ∨

4a − k

lem (with Neumann b.c.) min

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Heuristically we may expect for highly con- centrated sources a distribution of the low- congestion region around the sources. On the contrary, for sources with a low con- centration, we may expect a distribution of the low-congestion region mostly between f+ and f−. In the following examples, we consider f+ and f− two Gaussian distributions with vari- ance λ, centered at two points x0 and x1. We also take a = 1 and b = 4 (at equal traf- fic density the velocity in the low-congestion region = four times the one in the region with normal congestion).

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Gaussian sources (left) with variance λ = 0.02; plot of θ (right) using the penalization parameter k = 0.4. Computations made by Serena Guarino using FreeFem.

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Gaussian sources (left) with variance λ = 0.001; plot of θ (right) using the penalization parameter k = 0.01. Computations made by Serena Guarino using FreeFem.

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A free boundary problem arising in PDE

(Joint work with E. Oudet and B. Velichkov) In the problem above assume we have a ground congestion given by the function H(σ) =

resources produces a lower congestion like

sup

inf

2 |∇u|2 − fu

where the total amount of resources to spend is fixed.

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In this case the Dirichlet zero boundary con- dition means that we want to transport the mass f(x) dx to the boundary of D. A similar situation occurs when we have the Neumann boundary condition but for a right-hand side f with zero mean. Interchanging the sup and the inf above gives inf

sup

2 |∇u|2 − fu

and now the sup with respect to θ, for a fixed u, is easy to compute and we end up with min

2

2 ∇u2

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The existence of a solution ¯ u for this last problem is straightforward and, by strict con- vexity it is also unique. In order to solve the initial optimization problem the questions are to recover the optimal function ¯ θ from ¯ u and to describe the boundary of the free set ¯ Ω =

u| < ∇¯ u∞

f = 1. Indeed, we may show the equivalence with the obstacle problem min

2|∇u|2−u

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where d(x) is the distance function from ∂D and k = ∇¯ u∞. In this case the free set ¯ Ω =

u| < ∇¯ u∞

complement

Since the solution uk of the obstacle prob- lem is continuous, the free set ¯ Ω is open.

with the obstacle problem, we may conclude that the free boundary ∂ ¯ Ω is C1,α up to a singular set of zero Hausdorff d − 1 measure.

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that is the set where the distance function d is singular, is fully contained into the free set ¯ Ω.

explicit expression of ¯ θ can be computed: ¯ θ(r) =

am − 1

where am is a suitable constant.

for a general right-hand side f.

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In this case the optimal ¯ θ is given by (q = p′) ¯ θ = m1/p|∇u|2/(p−1)

where u solves the minimum problem for 1 2

2

−

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sumed regular, can we obtain regularity re- sults for the free boundary ∂ ¯ Ω? This would imply that on the free set ¯ Ω the PDE −∆¯ u = f

result for ¯ u.

Under which conditions on the data the free set ¯ Ω does not touch the exterior boundary ∂D? This seems to hap- pen in several numerical computations.

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function ¯ θ exists? In this case its support is contained in D \ ¯ Ω; moreover, if the free set ¯ Ω is regular, we obtain that ¯ θ satisfies the first order equation − div(¯ θ∇¯ u) = f in D \ ¯ Ω. In the torsion case this amounts to the PDE ∇¯ θ · ∇d + ¯ θ∆d = cf in D \ ¯ Ω for a suitable constant c.

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Plot of the gradient when D is the unit disk and f = 1. The red line is the free boundary, m = 0.5 left, m = 0.1 right.

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Plot of the gradient when D is the unit square and f = 1. The red line is the free boundary, m = 0.5 left, m = 0.1 right.

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Plot of the gradient when D is an ellipse and f = 1. The red line is the free boundary, m = 0.5 left, m = 0.1 right.

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Plot of the gradient when D is a treffle and f = 1. The red line is the free boundary, m = 0.5 left, m = 0.1 right.

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