Optimal regions for congested transport Giuseppe Buttazzo Dipartimento di Matematica Universit` a di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it Workshop “Optimal Transport in the Applied Sciences” Johann Radon Institute (RICAM) Linz, December 8–12, 2014
Joint work with: Guillaume Carlier - Paris Dauphine Serena Guarino - University of Pisa Math. Model. Numer. Anal. (M2AN), (to appear) available at: http://cvgmt.sns.it arxiv.org 1
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 of residents in Ω and f − the distribution of working places. We assume � � Ω f + dx = Ω f − dx. We also assume that the transport in Ω is congested and that the congestion function is given by a convex nonnegative superlinear function H . 2
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 Ω H ( σ ) dx : σ ∈ Γ f where � � Γ f = − div σ = f in Ω , σ · n = 0 on ∂ Ω , being f = f + − f − . See for instance Brasco- Carlier, Carlier-Jimenez-Santambrogio for de- tails on the model. 3
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 . 4
Our goal is to reduce the congestion acting on a suitable region C which has to be deter- mined. More precisely, two congestion func- tions H 1 and H 2 are given, with H 1 ≤ H 2 , 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. 5
For every region C we then consider the shape function � � � � F ( C ) = min Ω \ C H 2 ( σ ) dx + C H 1 ( σ ) dx : σ ∈ Γ f and so the optimal design of the low-congestion region amounts to the minimization problem � � min F ( C ) + m ( C ) : C ⊂ Ω . Much of the analysis below depends on the penalization function m ( C ). 6
The case m ( C ) = k Per( C ) Theorem For every k > 0 there exists an optimal solution C opt . Indeed, a minimizing sequence C n has a uni- formly bounded perimeter and so we may as- sume that C n tends strongly in L 1 to some The perimeter is L 1 -lower semicon- set C . tinuous; moreover, the optimal σ n ∈ Γ f pro- viding the value � � F ( C n ) = H 2 ( σ n ) dx + H 1 ( σ n ) dx Ω \ C n C n 7
are weakly L 1 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: ∇ H ∗ 1 ( ∇ u int ) in C σ = ∇ H ∗ 2 ( ∇ u ext ) in Ω \ C where u int and u ext solve the PDEs 8
� � ∇ H ∗ − div 1 ( ∇ u int ) = f in C ∇ H ∗ 1 ( ∇ u int ) · ν = 0 on ∂ Ω ∩ C � � ∇ H ∗ − div 2 ( ∇ u ext ) = f in Ω \ C ∇ H ∗ 2 ( ∇ u ext ) · ν = 0 on ∂ Ω ∩ (Ω \ C ) with the transmission condition across ∂C ∇ H ∗ 1 ( ∇ u int ) −∇ H ∗ 2 ( ∇ u ext ) · ν C = 0 on ∂C ∩ Ω . Performing the shape derivative on ∂C we also obtain on ∂C ∩ Ω 9
� ∇ H ∗ � � ∇ H ∗ � H 2 2 ( ∇ u ext ) − H 1 2 ( ∇ u ext ) ≤ k H C ∇ H ∗ ∇ H ∗ � � � � ≤ H 2 1 ( ∇ u int ) − H 1 1 ( ∇ u int ) . where H C is the mean curvature on ∂C . Since H 1 ≤ H 2 this gives that H C ≥ 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. 10
The case m ( C ) = k | C | Passing from sets C to density functions θ ( x ) ∈ [0 , 1] we obtain the relaxed formulation � � � � � min θH 1 ( σ ) + (1 − θ ) H 2 ( σ ) + kθ dx . σ,θ Ω The minimization with respect to θ is straight- forward and the optimal θ is θ = 1 H 1 ( σ )+ k<H 2 ( σ ) , which reduces the problem to � � � min Ω H 2 ( σ ) ∧ H 1 ( σ ) + k dx. σ 11
Since the function H = H 2 ∧ ( H 1 + k ) is not convex, a further relaxation gives finally the problem � Ω H ∗∗ ( σ ) dx. min σ If σ is a solution, we have • if H ∗∗ ( σ ) = H 2 ( σ ) we take θ = 0, that is no improvement for low congestion is needed; • if H ∗∗ ( σ ) = H 1 ( σ ) + k we take θ = 1, that is in this region we have to spend a lot to improve the congestion; • if H ∗∗ ( σ ) < H ( σ ) we have to spend re- sources proportionally to θ ( x ) ∈ ]0 , 1[. 12
If H 1 and H 2 only depend on | σ | we get 0 if | σ | ≤ r 1 | σ |− r 1 θ ( x ) = if r 1 < | σ | < r 2 r 2 − r 1 1 if | σ | ≥ r 2 where r 1 and r 2 can be explicitly computed from H 1 and H 2 : H ∗∗ ( r ) = H 2 ( r ) r 1 = max. solution of H ∗∗ ( r ) = H 1 ( r ) + k. r 2 = min. solution of Some numerical computations can be made when H 1 and H 2 are quadratic. 13
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: H 1 ( σ ) = a | σ | 2 , H 2 ( σ ) = b | σ | 2 with a < b. Then we have H ∗ ( ξ ) = ξ 2 � ξ 2 � 4 b ∨ 4 a − k and we simply have to solve the elliptic prob- lem (with Neumann b.c.) � � � � � H ∗ ( ∇ u ) − fu min dx . Ω 14
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 x 0 and x 1 . 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). 15
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 . 16
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 . 17
A free boundary problem arising in PDE optimization (Joint work with E. Oudet and B. Velichkov) In the problem above assume we have a ground congestion given by the function H ( σ ) = 2 | σ | 2 and that, investing an amount θ of 1 resources produces a lower congestion like 1 2(1+ θ ) | σ | 2 . We have then the problem � 1 + θ � � |∇ u | 2 − fu sup inf dx 2 u ∈ H 1 D � 0 ( D ) D θ dx = m where the total amount of resources to spend is fixed. 18
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 � 1 + θ � � |∇ u | 2 − fu inf sup dx 2 u ∈ H 1 D 0 ( D ) � D θ dx = m and now the sup with respect to θ , for a fixed u , is easy to compute and we end up with � 1 D |∇ u | 2 dx + m � � � 2 �∇ u � 2 min ∞ − D fu dx . 2 u ∈ H 1 0 ( D ) 19
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 � ∞ . • An easier case is the torsion problem, where f = 1. Indeed, we may show the equivalence with the obstacle problem � 1 � � � � 2 |∇ u | 2 − u dx : u ∈ H 1 min 0 ( D ) , u ( x ) ≤ kd ( x ) D 20
where d ( x ) is the distance function from ∂D and k = �∇ ¯ u � ∞ . In this case the free set � � ¯ Ω = |∇ ¯ u | < �∇ ¯ u � ∞ coincides with the � � complement of the contact set. | u | < kd Since the solution u k of the obstacle prob- lem is continuous, the free set ¯ Ω is open. • Still in the torsion case, by the equivalence with the obstacle problem, we may conclude Ω is C 1 ,α up to a that the free boundary ∂ ¯ singular set of zero Hausdorff d − 1 measure. 21
• Still in the torsion case, the cut locus of D , that is the set where the distance function d is singular, is fully contained into the free set ¯ Ω. • When f = 1 and D is the unit ball, the explicit expression of ¯ θ can be computed: � r � + ¯ θ ( r ) = − 1 a m where a m is a suitable constant. • The previous argument cannot be repeated for a general right-hand side f . 22
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