prob transport

Mass Transfer ProblemsOptimal NetworksPath Functionals

Optimization Problemsfor Transportation Networks

PhD Thesis

Alessio BrancoliniAdvisor: Prof. Giuseppe Buttazzo

Scuola Normale Superiore, Pisa

Thursday, October 12th 2006

Alessio Brancolini Optimization Problems for Transportation Networks


Outline

1 Optimal Transportation ProblemsMonge’s ProblemKantorovich’s ProblemComparison of the Problems

2 Optimal Networks for Mass Transportation ProblemsSetting of the ProblemTools and results

3 Path Functionals over Wasserstein SpacesPath Functionals over Wasserstein SpacesXia’s Transport PathsMaddalena, Morel, Solimini’s Irrigation Trees



Monge’s ProblemKantorovich’s ProblemComparison of the Problems

Mass Transfer Problems

Mass Transfer Problems




Statement of Monge’s Problem

Problem (Monge’s Problem, 1781)

X metric space; µ+, µ− finite positive Borel measures;mass balance condition µ+(X ) = µ−(X );

minimize

M(t) :=

∫X

c(x , t(x)) dµ+(x);

among measurable maps t : X → X such thatµ−(B) = µ+(t−1(B)) (transport maps); c : X × X → R,non-negative, l.s.c.




Statement of Kantorovich’s Problem

Problem (Kantorovich’s Problem, 1940)

X metric space; µ+, µ− finite positive Borel measures;mass balance condition µ+(X ) = µ−(X );

minimize

K (µ) :=

∫X×X

c(x , y) dµ(x , y);

among positive Borel measures on X × X such thatµ+(A) = µ(A× X ), µ−(B) = µ(X × B) (transport plans);c : X × X → R, non-negative, l.s.c.




Comparison of the Problems

if t is transport map, the measure µt given byµt(A× B) := µ+(A ∩ t−1(B)) is a transport plan andM(t) = K (µt).

Disadvantages of Monge’s Formulation:non-linearity of M in the unknown of the problem t ;there may be no transport maps.

Advantages of Kantorovich’s Formulation:linearity of K in the unknown of the problem µ;the set of transport plan is not empty, convex, weakly-∗

compact;the previous items imply the existence of an optimalmeasure.



Setting of the ProblemTools and results

Transportation Networks

Transportation Networks




A Model of Urban Optimization I

Ω

spt

spt

µ+

µ−

Σ




A Model of Urban Optimization II

γ

Σ

y

γ

x

Σ

dΣ(x , y) = A(H1(γ \ Σ)) + B(H1(γ ∩ Σ)) + C(H1(Σ))




A Model of Urban Optimization III

γ

Σ

y

γ

x

Σ

dΣ(x , y) = infA(H1(γ \ Σ)) + B(H1(γ ∩ Σ)) + C(H1(Σ)) :

γ closed connected, x , y ⊆ γ ⊆ Ω




A Model of Urban Optimization IV

µ+, µ− are the distributions of working people and workingplaces;

the “pseudo-distance” dΣ takes into account thetransportation network Σ via a function J;

in order to introduce the distance dΣ we consider afunction J : [0,+∞]3 → [0,+∞], e.g.J(a, b, c) = A(a) + B(b) + C(c);

the cost of the transportation on a given path γ, will then be

J(H1(γ \ Σ),H1(γ ∩ Σ),H1(Σ)).




A Model of Urban Optimization V

Ω ⊆ RN open bounded set, Σ ⊆ Ω closed connected set,µ+, µ− probability measures on Ω;

J : [0,∞]3 → R l.s.c., non-negative, non-decreasing,continuous in a; J(a, b, c) ≥ G(c) with G(c) → +∞ asc → +∞;

define

dΣ(x , y) = infJ(H1(γ \ Σ),H1(γ ∩ Σ),H1(Σ)) :

γ closed connected, x , y ⊆ γ ⊆ Ω;

T (Σ) is minimum of Kantorovich functional between themeasures µ+, µ− with cost function dΣ.




A Model of Urban Optimization VI

Problem

Minimize the functional

Σ 7→ T (Σ)

among Σ ⊂ Ω close connected sets.




A Model of Urban Optimization VII

(X , d) complete metric space.

Theorem (Gołab)

Cn closed connected sets, Cn → C. Then

H1(C) ≤ lim infn→∞

H1(Cn).

Theorem (Gołab, generalized)

Γn,Σn closed connected sets, Γn → Γ, Σn → Σ. Then

H1(Γ \ Σ) ≤ lim infn→∞

H1(Γn \ Σn).




A Model of Urban Optimization VIII

Theorem

J : [0,∞]3 → R satisfying the hypoteses before.

Then, there exists a minimizer of the functional T in the class ofclosed connected sets.



Path Functionals over Wasserstein SpacesXia’s Transport PathsMaddalena, Morel, Solimini’s Irrigation Trees

Tree-shaped Structures

Tree-shaped Structures




Tree-shaped Structures I

Many natural systems show a distinctive tree-shapedstructure: plants, trees, drainage networks, root systems,bronchial and cardiovascular systems.

These systems could be described in terms of masstransportation, but Monge-Kantorovich theory turns out tobe the wrong mathematical model since the mass iscarried from the initial to the final point on a straight line.




Tree-shaped Structures II

x1

x2

y1

x1

x2

y1

Figure: V-shaped versus Y-shaped transport.




Path Functionals

Path Functionals




Path Functionals, Setting of the Problem

(X , d) metric space such that bounded closed sets arecompact.

The functional we consider is

J (γ) =

∫ 1

0J(γ(t))|γ′|(t) dt ,

γ ∈ Lip([0, 1], X ) such that γ(0) = x0 and γ(1) = x1;

J : X → [0,+∞] is l.s.c.;

|γ′|(t) is the metric derivative, i.e.

|γ′|(t) = lims→t

d(γ(s), γ(t))|s − t |

.




Path Functionals, Existence Result

TheoremAssume that

there exists a curve γ0 such that J (γ0) < +∞,

and ∫ ∞

0

[inf

Br (x0)J]

dr = +∞ or J ≥ c > 0,

then the functional admits a minimum in the class of Lipschitzcurves γ : [0, 1] → X such that γ(0) = x0, γ(1) = x1.




Wasserstein Spaces

Wp(Ω), the p-Wasserstein space, the set of Borelprobability measures with finite momentum of order p, thatis all probability measures µ such that∫

X[d(x0, x)]p dµ(x) < +∞;

The p-Wasserstein distance between µ+, µ− is the givenby Kantorovich functional with cost c(x , y) = d(x , y)p

Wp(µ+, µ−) :=

[min

∫X×X

[d(x , y)]p dµ(x , y)

] 1p

.




Path functionals over Wasserstein Spaces

X = Wp(Ω); Ω compact metric space; m positive, finite,non-atomic Borel measure.

the coefficient J is l.s.c. local functional defined onmeasures given by:

J(µ) =

∫Ω

f(

dµ

dm

)dm +

∫Ω\Aµ

f∞(

dµs

d|µs|

)d|µs|+∫

Aµ

g(µ(x)) d#(x).

f : R → [0,+∞] convex, l.s.c. and proper; g : R → [0,+∞]l.s.c., subadditive, g(0) = 0; g0(s) = f∞(s).




Path Functionals, Existence Theorem in Wp(Ω)

Theorem

If f (s) > 0 for s > 0 and g(1) > 0, then J ≥ c > 0;

in particular, the functional

J (γ) =

∫ 1

0J(γ(t))|γ′|(t) dt

achieves a minimum in Lip([0, 1],Wp(Ω)) such thatγ(0) = µ+, γ(1) = µ−, provided there exists a curve offinite value.




Path Functionals, Concentration and Diffusion Cases

Ω compact, convex subset of RN .

Concentration case: f = +∞, g(z) = |z|r , (0 ≤ r < 1),

J(µ) = Gr (µ) =

∑k∈N(ak )r µ =

∑k∈N akδxk

+∞ otherwise;

Diffusion case: f (z) = |z|q, g = +∞, (q > 1),

J(µ) = Fq(µ) =

∫Ω |u|

qdx µ = u · LN

+∞ otherwise.




Path Functionals, Results on Concentration Case

if µ+, µ− are atomic measures, then there exists anoptimal curve γopt such that the functional is finite;

if r > 1− 1/N, the same is true for all couple of measures;

if r ≤ 1− 1/N, measures µ+, µ− such that the functional isnot finite on every curve connecting them can be found.




Path Functionals, Results on Diffusione Case

if µ+, µ− ∈ Lq(Ω), then there exists an optimal curve γopt

such that the functional is finite;

if q < 1 + 1/N, the same is true for all couple of measures;

if q ≥ 1 + 1/N, measures µ+, µ− such that the functional isnot finite on every curve connecting them can be found.




Transport Paths

Transport Paths




Transport Paths between Atomic Measures I

Ω ⊂ RN compact, convex;

µ+ =∑m

i=1 aiδxi , µ− =

∑nj=1 bjδyj convex combinations of

Dirac measures;

G weighted directed graph; spt µ+, spt µ− ⊆ V (G);

the mass flows from the initial measure µ+ to the finalmeasure µ− “inside” the edges of the graph G;

y1

y2

y3

x1

x2

x3




Transport Paths between Atomic Measures II

The point is now to provide to each transport path G asuitable cost that makes keeping the mass togethercheaper. The right cost function is

Mα(G) :=∑

e∈E(G)

[w(e)]αl(e),

l(e) length of edge e, 0 ≤ α < 1 fixed;

this cost takes advantage of the subadditivity of thefunction t 7→ tα in order to make the tree-shaped graphsmore economic.




Transport Paths between General Measures I

e (oriented edge) 7→ µe = (H1 e)e (vector measure),div µe = δe+ − δe− ;

G 7→ TG =∑

e∈E(G) w(e)µe;

div G = µ+ − µ− sums up all conditions;

a general transport path is defined by density and the costas a lower semicontinuous envelope:

Mα(T ) = infTGi

→Tlim infi→+∞

Mα(TGi).




Transport Paths, Some Results

if 1− 1/N < α ≤ 1, the existence of an optimal transportpath is assured between arbitrary measures;

conditions to link arbitrary measures have been found byDe Villanova, Solimini;




Irrigation Trees

Irrigation Trees




Irrigation Trees I

(Ω,M, µ) probability space;

associate to each ω ∈ Ω a 1-Lipschitz curveχω : [0,+∞[→ RN such that χω(0) = S for a.e. ω (set offibres);

[ω]t = ω′ : (χω′)|[0,t] = (χω)|[0,t];Mt(χ) = ω : χω not constant on [t ,+∞].




Irrigation Trees II

Definition (MMS functional)

Let α ∈ [0, 1]. The MMS functional is defined by

MMS(χ) :=

∫ +∞

0

[∫Mt (χ)

[µ([ω]t)]α−1 dµ(ω)

]dt .

Definition

A sequence of set of fibres χn is said to converge to a set offibres χ if, for a.e. ω, χω,n(t) → χω(t) for all t ∈ [0,+∞[.




Irrigation Trees, Some Results

Theorem

In a suitable subset functionals of the type

χ 7→ MMS(χ) + F (µχ),

with F is a l.s.c., admits a minimizer.

A case of interest is that of F finite on a certain measure µ−

and infinity elsewhere. Xia’s model and Maddalena, Morel,Solimini’s model come out to be the same.


Bibliography

Bibliography I

A. Brancolini and G. Buttazzo.Optimal networks for mass transportation problems.

A. Brancolini, G. Buttazzo, and F. Santambrogio.Path Functionals over Wasserstein Spaces.

G. Buttazzo, E. Oudet, and E. Stepanov.Optimal Transportation Problems with Free DirichletRegions.

F. Maddalena, J.M. Morel, and S. Solimini.A Variational Model of Irrigation Patterns.

Q. Xia.Optimal Paths Related to Transport Problems.


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