Sur un Problème de Monge de Transport Optimal de Masse · Sur un Probl eme de Monge de Transport...

Sur un Probleme de Monge de TransportOptimal de Masse

Noureddine Igbida

LAMFA CNRS-UMR 6140Universite de Picardie Jules Verne, 80000 Amiens

Limoges, 25 Mars 2010MODE 2010, Limoges

Noureddine Igbida Sur un Probleme de Monge de Transport Optimal de Masse

Monge problem

Y

f^+ f^−

t

=t#f^+

x y=t(x)

X

∫Ω

f +(x) dx =

∫Ω

f−(y) dy

A(f +, f−) =

t : X → Y ; µ+

#t = µ−, i.e.µ−(B) = µ

+(t−1(B))

where µ+ := f +(x) dx and µ− := f−(y) dy .

Monge problem : find a map T∗ ∈ A(f +, f−) which minimizes the cost functional

Fc (T∗) = mint∈A(f +,f−)

Fc (t) where Fc (t) =

∫X

c(x, t(x))f +(x) dx.

T∗ is called an optimal transport map pushing f + to f−.

Non existence :

d

X Y

µ^+ µ^−

e

a

b

c

µ+ = 2 δa + δb µ

− = δc + δd + δe


Monge-Kantorovich problem

µ^−

X

Y

µ

µ^+

µ+ = projxµ µ

− = projyµ

Consider

π(f +, f−) :=

µ ∈ Mb(X × Y ) ; µ+ = projxµ and µ− = projyµ

Monge-Kantorovich : find a measure µ∗ ∈ π(f +

, f−)

Kc (µ∗) := minµ∈π(f +,f−)

Kc (µ), where Kc (µ) =

∫c(x, y) dµ(x, y)

µ∗ is called an optimal transport plan between f + and f−.

Proposition

Let c : RN × RN → [0,+∞] be a lower semi-continuous cost function, and let f± ∈ L1(Ω) betwo non-negative Borel functions satisfying the mass balance condition. Then, there exists anoptimal transport plan µ∗ ∈ π(f +, f−) solving the Monge-Kantorovich problem.Moreover, we have

minKc (µ) : µ ∈ π(f +, f−) ≤ infFc (T ) : T ∈ A(f +

, f−).


Monge-Kantorovich problem

µ^−

X

Y

µ

µ^+

µ+ = projxµ µ

− = projyµ

Consider

π(f +, f−) :=

µ ∈ Mb(X × Y ) ; µ+ = projxµ and µ− = projyµ

Monge-Kantorovich : find a measure µ∗ ∈ π(f +

, f−)

Kc (µ∗) := minµ∈π(f +,f−)

Kc (µ), where Kc (µ) =

∫c(x, y) dµ(x, y)

µ∗ is called an optimal transport plan between f + and f−.

Proposition

Let c : RN × RN → [0,+∞] be a lower semi-continuous cost function, and let f± ∈ L1(Ω) betwo non-negative Borel functions satisfying the mass balance condition. Then, there exists anoptimal transport plan µ∗ ∈ π(f +, f−) solving the Monge-Kantorovich problem.Moreover, if c is continuous, then

minKc (µ) : µ ∈ π(f +, f−) = infFc (T ) : T ∈ A(f +

, f−).


Kantorovich dual problem

f^+f^−

Y

X

f +(x)u(x) ...... f−(x)v(y)u(x) + v(y) ≤ c(x, y)

Φc (f +, f−) :=

(u, v) ∈ L1(Ω)× L1(Ω) : u(x) + v(y) ≤ c(x, y)

X ∪ Y ⊆ Ω a bounded domain of RN

Kantorovich dual problem : find (u∗, v∗) that maximize

D(u∗, v∗) = max(u,v)∈Φc (f +,f−)

D(u, v) where D(u, v) :=

∫X

u f + +

∫Y

v f−

We have the following result (see for instance [Ambrosio-Pratelli] or [Villani]).

Theorem (Kantorovich duality)

Let c : RN × RN → [0,+∞] be a lower semi-continuous cost function, and let f± ∈ L1(Ω) betwo non-negative Borel functions satisfying the mass balance condition. Then,

minKc (µ) : µ ∈ π(f +, f−) = supD(u, v) : (u, v) ∈ Φc (f +

, f−).


Which cost functions are of interest ?

general cost c l.s.c. : Ambrosio, Pratteli, Villani, Gangbo, McCann, ....

Euclidienne cost c(x, y) = |x − y | : classical Monge problem, mass optimization (Bouchitte,Buttazzo ...), granular matter (Evans, Caffarelli, McCann, Prigozhin, Barret, Igbida ...)

c(x, y) = |x − y |2 : meteorology (Cullen, Norbury, Purser, Shutts ...), hydrodynamic (Brenier...), statistics (Knott, Smith ...), Steepest descent or gradient descent method for PDE ...

c(x − y) = |x − y |p for 0 < p < 1 (or generally a strictly increasing and strictly concavefunctions of distance) : realistic cost in economy

c(x − y) = |x − y |p for 1 < p <∞, probability and statistic (Rachev ....)

strictly concave or convex functions of distance : Gangbo, McCann ...

Our aim :

To fix the length of a step and the distance that measures the cost of the transport dependsof the number of steps that is needed to transport the masses.

The limit as the length tends to 0


Our aim : discrete distance cost function

Discrete distance for step of size ε :

dε(x, y) =

0 if x = y ,

ε([[|x−y|ε

]]+ 1)

if x 6= y .

where |.| is the Euclidean norm and [[r ]] is defined for r > 0 by

[[r ]] := n, if n < r ≤ n + 1, n = 0, 1, 2, . . . .

x

f^+ f^−

epsilon

t_1

t_4

t_2t_3

dε(x, t1(x) = dε(x, t2(x) = ... = dε(x, t4(x)).

Example

For ε = 1, we obtain

d1(x, y) =

0 if x = y ,1 if 0 < |x − y | ≤ 1,2 if 1 < |x − y | ≤ 2,

.

.

.


Kantorovich-Rubinstein Theorem (metric cost function)

Theorem (Kantorovich-Rubinstein Theorem)

If c = d is a lower semi-continuous metric on RN and f± ∈ L1(Ω) are two non-negative Borelfunctions satisfying the mass balance condition, then

minKd (µ) : µ ∈ π(f +, f−) = maxPf +,f− (u) : u ∈ Kd (Ω)

,

where Pf +,f− (u) :=

∫Ω

u(x)(f +(x)− f−(x)) dx and

Kd (Ω) := Lip1(Ω, d) =

u ∈ L1(Ω) : |u(x)− u(y)| ≤ d(x, y) a.e. in Ω.

The maximizers u∗ are called Kantorovich (transport) potentials. Moreover, for an optimal

transport map T∗ pushing f + to f−, we have (Dual Criteria for Optimality)

u∗(x)− u∗(T∗(x)) = d(x,T∗(x)) for almost all x ∈ supp(f +),

being u∗ a Kantorovich potential.

The Kantorovich transport potentials are solution of the

f + − f− ∈ ∂IKd (Ω)(u),

where

IKd(u) :=

0 if u ∈ Kd ,+∞ if u 6∈ Kd ,

i.e. u is a solution of

max

∫Ω

u(x)(f +(x)− f−(x)) dx : u ∈ Lip1(Ω, d)

.


Euclidean metric : Evans-Gangbo PDE (Monge-Kantorovich equation)

Assumec(x, y) = |x − y |.

Theorem (Evans-Gangbo Theorem)

Let d|·| be the Euclidean distance on RN , and let f± ∈ L1(Ω) be two non-negative Borel function

satisfying the mass balance condition. Assume additionally that f± are Lipschitz continuousfunctions with compact support such that supp(f +)∩ supp(f−) = ∅. Then, there exists u∗ ∈ K|·|such that∫

Ω

u∗(x)(f +(x)− f−(x)) dx = max

∫Ω

u(x)(f +(x)− f−(x)) dx : u ∈ K|·|

;

and there exists 0 ≤ a ∈ L∞(Ω) such thatf + − f− = −div(a∇u∗) in D′(Ω)a (|∇u∗| − 1) = 0.

Monge-Kantorovich equation is the Euler Lagrange equation for the dual Kantorovichproblem.

The function a is the Lagrange multiplier corresponding to the constraint |∇u∗| ≤ 1, and itis called the transport density. ∇u gives the direction of the transport.

The optimal map transport T∗ is given by the solution of the following ODE : z′(t) =−a(z(t))∇u∗(z(t))

t f−(z(t)) + (1− t) f +(z(t))for t ∈ [0, 1]

z(0) = z0 ∈ X ,

thenT∗(z0) = z(1) for any z0 ∈ X .


Preliminary results for the discrete distance cost

Since the cost is lower semi-continuous :

Corollary

For any f± ∈ L1(Ω) satisfying the mass balance condition, we have

there exists an optimal plan transport µ∗ minimizing the Monge-Kantorovich problem

Kdε (µ∗) = minKdε (µ) : µ ∈ π(f +, f−).

The Kantorocich duality remains true, i.e.,

minKdε (µ) : µ ∈ π(f +, f−) = supD(u, v) : (u, v) ∈ Φdε (f +

, f−).

minKdε (µ) : µ ∈ π(f +, f−) ≤ inf

Fdε (T ) : T ∈ A(f +

, f−).

Since the cost is a metric (by Kantorovich-Rubinstein Theorem) :

Corollary

For any f± ∈ L1(Ω) satisfying the mass balance condition, we have

minKdε (µ) : µ ∈ π(f +, f−) = maxPf +,f− (u) : u ∈ Kdε (Ω)

.

Moreover, for an optimal transport map T∗ pushing f + to f−, we have (the Dual Criteria forOptimality)

u∗(x)− u∗(T∗(x)) = dε(x,T∗(x)) for almost all x ∈ supp(f +),



Equality between Monge’s infimun and Kantorovich’s minimum

The same example given by Pratelli proves that the equality between Monge’s infimun andKantorovich’s minimum is not true for the case of the cost function given by the metric d1 :

Example

R := (−1, y) : y ∈ [−1, 1], S := (0, y) : y ∈ [−1, 1], Q := (1, y) : y ∈ [−1, 1].

Q

0

1

−1

1

S

−1 x

y

R

f + := 2H1 S f− := H1 R +H1 Q

Then

minKd1(µ) : µ ∈ π(f +

, f−) = 2 < 4 ≤ infFd1

(T ) : T ∈ A(f +, f−)

.

The cause here is not the discontinuity of d1, but is the fact that the measures f + and f− are notabsolutely continuous respect the Lebesgue measure.

Theorem (Ig-Mazon-Toledo-Rossi,2010)

Let f± ∈ L1(Ω) be two non-negative Borel functions satisfying the mass balance condition. Then,

minKdε (µ) : µ ∈ π(f +, f−) = inf

Fdε (T ) : T ∈ A(f +

, f−).


Characterization of the Kantorovich potential

Theorem (Ig-Mazon-Toledo-Rossi,2010)

Let f± ∈ L∞(Ω) be two non-negative Borel functions satisfying the mass balance condition.

There exists a Kantorovich potential u∗ such that u∗(Ω) ⊂ Z and takes a finite number of

values. More precisely, there exists a partition Ajkj=0 of Ω such that

u∗ =k∑

j=0

j χAj.

In addition, we have f + χA0

= f− χAk= 0.

u∗ is a Kantorovich potential if and only if u∗ ∈ Kε and there exists

σ∗ ∈ Ma

b(Ω× Ω) := bounded antisimetric Radon measures in Ω× Ω ,

such thatσ∗ (x, y) ∈ Ω× Ω : |u∗(x)− u∗(y)| = ε, |x − y | ≤ ε,

projxσ∗ = f + − f− =: f and |σ∗|(Ω× Ω) =

2

ε

∫Ω

f (x) u∗(x) dx.

Moreover, [σ∗]+ (x, y) ∈ Ω× Ω : u∗(x)− u∗(y) = ε, |x − y | ≤ ε,

[σ∗]− (x, y) ∈ Ω× Ω : u∗(y)− u∗(x) = ε, |x − y | ≤ ε.


Nonexistence of an optimal transport map

There are several examples that show that the existence of a transport map is a delicate issue thatdepends strongly on the configuration of the data f + and f− and the distance d1.

Example

we prov ethat

0 2

1

L

1f^+ f^−

x

y

1

0 1 2 3

2

u^*

L=3

f + = L χ[0,1] f− = χ

[−L,0] u∗

By contradiction, we prove that there is no optimal map transport to push f + in f−, with d1.First step

The Dual Criteria for Optimality implies that : d1(x,T∗(x)) ∈ 1, 2

Consider Ai =

x ∈ [0, 1] ; d1(x,T∗(x)) = i, for i = 1, 2.

Again the Dual Criteria for Optimality implies that

T∗(A1) = [1, 2] and T∗(A2) = [2, 3]

Second step :

Using again u∗ and the Dual Criteria for Optimality, we prove that

T∗(x) = x + i forx ∈ Ai , i = 1, 2.

Contradiction : |Ai | = |T (Ai )|, for i = 1, 2.


Construction of an optimal plan transport

Let [σ∗]+ be the positive part of σ∗, given byσ∗ ∈ Ma

b(Ω× Ω), σ∗ (x, y) ∈ Ω× Ω : |u∗(x)− u∗(y)| = ε, |x − y | ≤ ε

projxσ∗ = f + − f− =: f and |σ∗|(Ω× Ω) =

2

ε

∫Ω

f (x) u∗(x) dx,


Recall that [σ∗]+ (x, y) ∈ Ω× Ω : u∗(x)− u∗(y) = ε, |x − y | ≤ ε,

[σ∗]− (x, y) ∈ Ω× Ω : u∗(y)− u∗(x) = ε, |x − y | ≤ ε.

Example (k = 3)

support(sigma^*−)

A_3 A_2 A_0A_1

u^*+

A_0

A_0A_3

A_3

A_2

A_2

A_1

A_1

support(sigma^*+)



Theorem

Let f± ∈ L∞(Ω) be satisfying the balance condition, u∗ =k∑

j=0

j χAjbe the Kantorovich potentiel

and [σ∗]+ be the measure given by the preceeding Theorem. Then,

[σ∗]+ =k∑

j=0

σ∗+j where

σ∗+j = [σ∗]+ Aj × Aj−1.

For any j = k, ..., 1, the measure σ∗+j is an optimal plan transport of

f +χAj

+ g χAjinto f− χAj−1

+ g χAj−1,

where g χAK

= g χA0= 0

g = projx (σ+)− f + = projy (σ+)− f−.

In other words, the sequence σ∗+j gives the following optimal transportation :

[σ∗]+ Ak × Ak−1 : f +χAk

into f− χAk−1+ g χAk−1

[σ∗]+ Ak−1 × Ak−2 : f +χAk−1

+ g χAk−1into f− χAk−2

+ g χAk−2

.

.

[σ∗]+ A1 × A0 : f +χA1

+ g χA1into f− χA0



Example (k = 3)

f^−

A_3 A_2 A_0A_1

u^*detailled transportation by sigma^*+

A_3

A_2

A_1

A_0

f^+

+g

f^+

+g

f^+

sigma^*+_3

sigma^*+_2

sigma^*+_1

f^−

f^−

How to construct an optimal plan transport of f + to f− ?

Lemma (Gluing Lemma, Villani)

Let f1, f2, h be three positive measures in Ω. If µ1 ∈ π(f1, h) and µ2 ∈ π(h, f2), there exits ameasure G(µ1, µ2) ∈ π(f1, f2) such that

Kd (G(µ1, µ2)) ≤ Kd (µ1) +Kd (µ2).



Example (k=3)

First step

[σ∗]+ A3 × A2 : f +

χA3to f− χA2

+ g χA2

[σ∗]+ A2 × A1 : f +χA2

+ g χA2to f− χA1

+ g χA1

[σ∗]+ A3 × A2 : f +

χA3f− χA2

+ g χA2

+f +χA2

δ[y=x] : +f +χA2

to +f +χA2

[σ∗]+ A2 × A1 : f +χA2

+ g χA2f− χA1

+ g χA1

+f− χA2δ[y=x] : +f− χA2

to +f− χA2

Gluing Lemma

ν2 := G(

[σ∗]+ A3 × A2 + f +χA2

δ[y=x], [σ∗]+ A2 × A1 + f− χA2δ[y=x]

)satisfies

ν2 : f +χA3

+ f +χA2

to f− χA1+ f− χA2

+ g χA1



Example (k=3)

Second and final step

ν2 : f +

χA3+ f +

χA2to f− χA1

+ f− χA2+ g χA1

[σ∗]+ A2 × A1 : f +χA1

+ g χA1to f− χA0

ν2 : f +

χA3+ f +

χA2f− χA1

+ f− χA2+ g χA1

+f +χA1

δ[y=x] : +f +χA1

to +f +χA1

[σ∗]+ A2 × A1 : f +χA1

+ g χA1f− χA0

+(f− χA1+ f− χA2

) δ[y=x] : +f− χA1+ f− χA2

to +f− χA1+ f− χA2

Gluing Lemma

ν1 := G(ν2 + f +

χA1δ[y=x], [σ∗]+ A2 × A1 + (f− χA1

+ f− χA2) δ[y=x]

)satisfies

ν1 : f +χA3

+ f +χA2

+ f +χA1

to f− χA0+ f− χA1

+ f− χA2



Example (k = 3)

=f^+

A_3

A_2

A_0

f^+

+g

f^+

+g

f^+

sigma^*+_3

sigma^*+_2

sigma^*+_1

f^−

f^−

f^−

+...

+...

A_1

g

f^+

f^−

f^+

f^−

A_3 A_2 A_1

A_2 A_1 A_0

A_3 A_2

A_2 A_1

nu_2 nu_2+...

nu_1

sigma^*+_1 + ....

fiest steplast step

glu

ein

g

glu

ein

g

=f^−

ν2 := G(

[σ∗]+ A3 × A2 + f +χA2

δ[y=x], [σ∗]+ A2 × A1 + f− χA2δ[y=x]

)ν1 := G

(ν2 + f +

χA1δ[y=x], [σ∗]+ A2 × A1 + (f− χA1

+ f− χA2) δ[y=x]

)Noureddine Igbida Sur un Probleme de Monge de Transport Optimal de Masse


The general case :

Now, for any j = k, ...1, let us consider

F(+)j :=

j∑i=k

f +χAi

and F(−)j :=

j−1∑i=k−1

f− χAi+ g χAj−1

.

Remark thatF

(+)1 = f + and F

(−)1 = f−

Let us consider the sequence of measures(νj

)j=k,..1

, defined by

νk = σ+ Ak × Ak−1

νj−1 = G(νj + f +

χAj−1δ[y=x], σ

+ Aj−1 × Aj−2 +

k−1∑i=j−1

f− χAiδ[y=x]

),

for j = k, ...1,

where G is the Gluing operator.

Theorem

For any j = k, ...1, νj is an optimal plan transport of F(+)j into F

(−)j . In particular ν1 is the

optimal mass transport of f + into f−.


Related works

http://www.mathinfo.u-picardie.fr/igbida/

N. Igbida, J. Mazon, J. Rossi et J. Toledo, A Nonlocal Monge-Kantorovich Problem,submitted.

N. Igbida, Partial Integro-Differential Equation in Granular Matter, submitted.

Find (u, µ) such that

projx

((u(x)− u(y)) µ(x, y)

)= f −

∂u

∂tin R2

|u(x)− u(y)| ≤ δ for |x − y | ≤ εµ > 0, Support(µ) ⊆

|u(x)− u(y)| = δ for |x − y | ≤ ε

F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo. The limit as p →∞ in a nonlocalp−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles. Calc.Var. 35 (2009), 279–316. Find (u, µ) such that

∫RN

J(x−y)|u(t, x)−u(t, y)|p−2(u(t, x)−u(t, y))dy = f (t, x)−∂u(t, x)

∂tfor (t, x) ∈ (0,T )×RN

.

N. Igbida and F. Karami, Numerical Analysis of a PIDE for Sandpile, in perparation.


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