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Sur l’equation de Monge-Kantorovich
Noureddine Igbida
LAMFA CNRS-UMR 6140,Universite de Picardie Jules Verne, 80000 Amiens
Le Havre, 16 Avril 2009
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 1 / 34
Main of the talk
Monge-Kantorovich evolution equation
Let Ω ⊂ RN be a bounded regular domain, N > 1, T > 0, Q = Ω × (0, T ) and Σ = ∂Ω × (0, T )
(EMK)
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:
∂h
∂t(x , t) −∇ · m(x , t)∇h(x , t) = µ in Q
m(x , t) > 0, |∇h(x , t)| ≤ 1 in Q
m (1 − |∇h(x , t)|) = 0 in Q
u = 0 on Σ
u(0) = u0 in Ω
Main interest :
1 Existence and uniqueness of a solution : µ a Radon measure
2 Numerical analysis : representation of t ∈ [0,T ) → (u(t, x), m(t, x) ∇u(t, x)).
3 Large time behavior, as t → ∞.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 2 / 34
Motivation
Motivation
Optimal mass transportation (Monge-Kantorovich problem) :- L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass
transfer problem. Mem. Am. Math. Soc., 137 (1999), no. 653.
Sandpile model :- L. Prigozhin, Variational model of sandpile growth. Euro. J. Appl. Math. , 7 (1996), 225-236.- G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diffusion and growing sandpiles. J. Differential
Equations, 131 :304-335, 1996.
Cellular Automaton :- L. C. Evans and F. Rezakhanlou. A stochastic model for sandpiles and its continum limit. Comm.
Math. Phys., 197 (1998), no. 2, 325-345.
Mass optimization :- G. Bouchitte and G. Buttazzo, Characterization of optimal shapes and masses through Monge-
Kantorovich equation. J. Eur. Math. Soc. 3 (2001), no. 2, 139-168.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 3 / 34
Motivation
Motivation I : optimal mass transportation
Monge optimal mass transfer [Monge-1780] : Given two measures µ+ and µ− in RN , such that
µ+(RN ) = µ−(RN ) and spt(µ+) 6= spt(µ−)
Mint∈A
Z
RN|x − t(x)| dµ+(x)
A =
t : spt(µ+) → spt(µ−) ; µ+#t = µ−
i.e. µ−(B) = µ+(t−1(B))
ff
˛
˛
˛
˛
˛
˛
˛
˛
˛
˛
x
t(x)mu^+
mu^−
t
Monge-Kantorovich problem [Kantorovich-1940] :
Realaxed variant : optimal plan transport
µ ∈ Π(µ+, µ−) :=n
γ ∈ Mb(IRN × IRN ), projx (γ) = µ+, projy (γ) = µ−
o
J (µ) = minγ∈Π(µ+,µ−)
J (γ) J (γ) =
Z Z
IRN×IRN|x − y | dγ(x , y)
Is µ supported in a graph ? i.e. ∃ ? t ∈ A, µ(E) = µ+(x ∈ IR
N; (x, t(x)) ∈ E)
Dual variational principle : Shipper (principe du convoyeur) µ is optimal if and only if
J (µ) = maxξ∈Lip1(RN )
n
Z
IRNξ dµ+ +
Z
IRNξ dµ−
o
=
Z
IRNu dµ+ +
Z
IRNu dµ− .
• Memoire sur la theorie de Deblais et des remblais, Gaspard Monge, Histoire de l’academie des Sciences de Paris, 1781.• On the transfer of massesm, L.V. Kantorovich, Dokl. Acad. Nauk. SSSR 37, 227-229 (1942).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 4 / 34
Motivation
Motivation I : optimal mass transportation
Monge optimal mass transfer [Monge-1780] : Given two measures µ+ and µ− in RN , such that
µ+(RN ) = µ−(RN ) and spt(µ+) 6= spt(µ−)
Mint∈A
Z
RN|x − t(x)| dµ+(x)
A =
t : spt(µ+) → spt(µ−) ; µ+#t = µ−
i.e. µ−(B) = µ+(t−1(B))
ff
˛
˛
˛
˛
˛
˛
˛
˛
˛
˛
x
t(x)mu^+
mu^−
t
Evans and Gangbo [1999] : How to fashion an optimal transport t ?
Monge-Kantorovich equation (Stationary equation of (EMK))
(MK)
−∇ · (m ∇u) = µ+ − µ−
m > 0, |∇u| ≤ 1, m(|∇u| − 1) = 0
ff
=⇒ m ∇u : the flux transportation
m gives the transport density−∇u gives the direction of the optimal transportation
Remark : If (u, m) solves (MK) then u is the Kantorovich potential ; i.e.
maxξ∈Lip1(RN )
Z
IRNξ dµ =
Z
IRNu dµ
• Memoire sur la theorie de Deblais et des remblais, Gaspard Monge, Histoire de l’academie des Sciences de Paris, 1781.Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 5 / 34
Motivation
Motivation II : growing sandpile model
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 6 / 34
Motivation
Motivation II : growing sandpile model
x
h(x,t)
f(x,t)
q(x,t)
Conservation of mass :The flow of the granular material is confined in a thin boundary layer moving down the slopes of agrowing pileThe density of the material is constant
∂h
∂t(x , t) = −∇ · q(x , t) + µ(x , t)
Surface flow directed by the steepest descent :
∃ m = m(x , t) > 0 such that q(x , t) = −m(x , t) ∇h(x , t)
No pouring over the parts of the pile surface inclined less than α :
|∇h(x , t)| < γ =⇒ m(x , t) = 0
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 7 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
X(t5)
0 1 0 0
0
0
0
0
0
f(t1) f(t2) f(t3) f(t4)
f(t5)
f(t5)0
X(0) X(t1) X(t2) X(t3)
X(t4)
X(t4)
X(t5)
X(t5)
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Adding a new cube on an existing pile, we have the following possibilities
the new cube remains in place
the new cube moves by falling in one of the forth directions (up, down, left or right).
there are several downhill ”staircases” along which the cube can move, and the cube will randomly selectamong the allowable downhill paths.
New cubes will move (or not) in order to get a stable configuartion, which means that the heights of any twoadjacent columns of cubes can differ by at most one :
|η(i) − η(j)| ≤ 1 if i ∼ j,
where η(i) denotes the height of the columns situated at the position i ∈ Z2.
• A stochastic model for growing sandpiles and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998), no2, 325-345.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 8 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Let p(i , j , η) : probability that a cube placed at the position i will end up at jSo, for any i , j ∈ Z2 we have
0 ≤ p(i , j , ξ) ≤ 1 andX
j∈Z2
p(i , j , ξ) = 1.
Adding cubes with a rate f , we have
IE [η(tk+1, j) | η(tk )] = η(tk , j) + c(j , η, tk ), for any j ∈ Z2.
c(j , η, τ) : (highly nonlocal factor) records the rate, at time τ, new cubes come to rest at j
c(j , η, t) =X
i∈Z2
p(i , j , η(t)) f (i , t), for any (j , t) ∈ Z2 × [0,∞).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 9 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Stochastic Equation (integral form) :0
B
B
@
∀ F : S × (0,∞) → R Lipchitz continuous in t and F (η(., 0), 0) = 0
IEh
F (η(., t), t) −
Z t
0
„
∂F
∂s+ Ls F
«
(η(., s))i
= 0,
where(Lt F )(ξ) :=
X
j∈ZN
c(j , ξ, t)“
F (Tj (ξ)) − F (ξ)”
.
with
Tj : ξ ∈ S → Tj (ξ) ∈ S with Tj (ξ) =
ξ(i) + 1 if i ∼ jξ(i) otherwise
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 10 / 34
Motivation
Motivation III : Sandpile Cellular Automaton
cubes distribution : mu
i ∼ j ⇐⇒ |i − j | ≤ 1 3D 2D
Assume that f ∈ BV (0, T ;L2(R2)
f (t, i) =1
Nf
„
t
N,
i
N
«
(η(t), t > 0) the associated Markov processusu the solution de (EMK)
Theorem
As N → ∞, we have
IE
"
supx∈R2
˛
˛
˛u(t, x) −1
Nη(t, [N x ])
˛
˛
˛
#
→ 0.
• A stochastic model for growing sandpiles and its continuum limit, L. C. Evans, F. Rezakhanlou, Comm. Math. Phys., 197 (1998), no2, 325-345.• Back on Stochastic Model for Sandpile, N. Igbida (soumis).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 11 / 34
Motivation
Boundary condition : Dirichlet
If µ+(IRN ) = µ−(IRN ) and µ+, µ− are supported in bounded domain : by taking Ω largeenough we can assume that u = 0 on ∂Ω;
If µ+(IRN ) 6= µ−(IRN ) and µ+, µ− are supported in bounded domain, then one needs :
There exists Γ such that HN−1(Γ) = 0 and replace the euclidean c(x, y) = |x − y| by thesemi-geodesic distance taking into account the boundary ; i.e.
cΓ(x, y) = min“
|x − y|, dist(x, Γ) + dist(y, Γ)”
.
Dirichlet Monge-Kantorovich problem (Bouchite, Buttazzo, De Pascale ...)
infn
Z Z
cΓ(x, y) dν(x, y) ; ν ∈ M(Ω × Ω) s.t. π1#ν − π
2#ν = µ1 − µ2
o
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 12 / 34
Main interest
Monge-Kantorovich evolution equation with Dirichlet boundarycondition
(EMK)
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:
∂h
∂t−∇ · (m ∇h) = µ in Q
m > 0, |∇h| ≤ 1 in Q
m (1 − |∇h|) = 0 in Q
u = 0 on Σ
u(0) = u0 in Ω
Main interest :
1 Existence and uniqueness of a solution : µ a Radon measure
2 Numerical analysis : representation of t ∈ [0,T ) → (u(t, x), m(t, x) ∇u(t, x))..
3 Large time behavior, as t → ∞.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 13 / 34
References
References de l’expose
http ://www.mathinfo.u-picardie.fr/igbida/
S. Dumont and N. Igbida,Back on a Dual Formulation for the growing Sandpile Problem, European Journal of AppliedMathematics, vol. 20, pp. 169-185, 2008.
N. Igbida,Evolution Monge-Kantorovich Equation,en revision dans Calculus of Variations and PartialDifferential Equations.
N. Igbida,On Monge-Kantorovich Equation, sous presse dans Nonlinear Analysis TMA.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 14 / 34
Difficulties and definitions of a solution
Main Difficulties
If m is regular then
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∂tu −∇ · (m ∇u) = µ
|∇u| ≤ 1, m > 0, m (|∇u| − 1) = 0
u = 0 on ∂Ω
⇒
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ut + ∂IIK (u) ∋ µ,
K =n
z ∈ H10 (Ω) ; |z | ≤ 1 in Ω
o
i.e. ∀ t ∈ [0,T ), u(t) ∈ K and
Z
Ωu(t)
“
µ(t) − ∂tu(t)”
= maxξ∈K
Z
“
µ(t) − ∂tu(t)”
ξ.
Nonlinear semi-group theory ⇒ existence and uniqueness of ”variational” solution u
Main Difficulties
The main information are in the flux Φ = m ∇u :
Monge-Kantorovich −→ flux transportSandpile −→ numerical analysis
In general, m is not regular : m is a measure.
How to define the solution for (MK) and (EMK) : m is a measure and ∇u ∈ L∞(Ω).
How to do numerical analysis for (MK) and (EMK) : regularisation technics are not stable.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 15 / 34
Difficulties and definitions of a solution
Notion of solution : how to define m ∇u (m a Radon measure)
Stationary problem : Theory of tangentiel gradient of u with respect to Radon measure (cf.Bouchitte, Buttazzo, De Pascale, Seppecher ... ) :
(MK)
−∇ · (m ∇mu) = µ in D′(Ω)|∇mu| = 1 m − a.e. in Ω.
(Monge-Kantorovich equation)
In particular :
Let ν ∈ Mb(Ω) and 1 < p < ∞
w = ∇νu ∈ Lpν(Ω) ⇔ ∃ uh ∈ C1(Ω), (uh,∇uh) → (u, w) in Lp
ν(Ω)N+1.
For any F ∈ Lp′
ν (Ω)N , such that −∇ · (F ) = µ ∈ Lp(Ω), we have
Z
Ω∇νu · F dν = −
Z
Ωu µ
for any u ∈ C0(Ω) such that u is Lipchitz.
Existence, uniqueness of m ∇mu, regularity, related transport for Monge problem... : Ambrosio,Bouchitte, Buttazzo, De Pascale, Evans, Seppecher ....
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 16 / 34
Difficulties and definitions of a solution
Notion of solution : how to define m ∇u (m a Radon measure)
Evolution problem :
Theory ? tangentiel derivative for time dependent functions : t ∈ (0, T ) → ∇νu(t).
For µ ∈ L2(Q) : existence, uniqueness and numerical analysis by dual formulation
Prigozhin - 1996 : m ∈“
L∞(Q)”∗
?
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Z Z
Q
(ut − µ) ξ +D
m,∇u · ∇ξE
= 0
D
m, ξE
> 0,D
m, |∇u|2 − 1 >= 0
Prigozhin and Barrett - 2006 : m ∇u =: Φ ∈ Mb(Q)N, ∇ · Φ ∈ L
2(Q).
Minimizing problem related to Φ and
Z
t
0Φ in Mb(Q)
Ngives Φ and u(t) = u0 +
Z
t
0f (s) ds + ∇ ·
Z
t
0Φ.
Equivalent formulation8
<
:
ut − ∇ · Φ = µ in D′(Q)
|Φ|(Q) =
Z Z
Qu ∇ · Φ
Our aim : To treat existence and uniqueness for general data (measures), large time behaviorand numerical analysis.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 17 / 34
Main results
Notion of solution : equivalent formulations
Remark
Assume that Φ : Ω → RN and u : Ω → R are regular and |∇u| ≤ 1 in Ω. Then
~
w
w
w
w
w
• m (|∇u| − 1) = 0 and Φ = m ∇u in Ω• m = |Φ| = Φ · ∇u in Ω
• m = |Φ| and
Z
Ω|Φ| ≤
Z
ΩΦ · ∇u
Theorem (Ig,2007)
Let µ be a Radon measure and u ∈ K . Then,Z
Ωu dµ = max
ξ∈K
Z
Ωξ dµ ⇔ ∃Φ ∈ Mb(Ω)N
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−∇ · (Φ) = µ in D′(Ω)
|Φ|(Ω) ≤
Z
Ωu dµ
(weak solution)
⇐⇒
−∇ · (Φ) = µ in D′(Ω)Φ = |Φ| ∇|Φ|u
⇐⇒
Z
Ωu dµ = |Φ|(Ω) = min|ν|(Ω) ; −∇ · ν = µ.
Remark
If (u, Φ) is a weak solution and Φ ∈ L1(Ω)N , then Φ = |Φ| ∇u a.e. in Ω.
The expression |Φ|(Ω) =
Z
Ωu dµ =
D
∇ · Φ, uE
is a weak formulation of |Φ| = Φ · ∇u.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 18 / 34
Main results
Notion of solution : Evolution Monge-Kantorovich equation
Lq(0,T ; w∗-Mb(Ω)) =n
ψ : (0,T ) → Mb(Ω), weak∗measurable ;
Z
T
0
|ψ(t)|q(Ω) <∞o
.
BV (0,T ; w∗-Mb(Ω)) =n
µ ∈ L1(0,T ; w∗-Mb(Ω)) ; lim suph→0
1
h
Z
T−h
0
|µ(τ+h)−µ(τ )|(Ω)dτ <∞o
.
In Q the condition |Φ| = Φ · ∇u mays be represented by one of the following formulation :
|Φ(t)|(Ω) ≤D
∇ · Φ(t), u(t)E
=D
µ(t) − ∂tu(t), u(t)E
for t ∈ (0,T )
|Φ|(Q) ≤
Z T
0
D
∇ · Φ, uE
=
Z T
0
D
µ(t) − ∂tu(t), u(t)E
Proposition (Ig,2007)
If Φ ∈ L1(0, T ;w∗-Mb(Ω))N and ∇ · Φ ∈ L1(0, T ; w∗-Mb(Ω)), u ∈ C([0, T ); L2(Ω)) ∩ KT ,
u(0) = u0, ∂tu −∇ · Φ = µ in D′(Q), then
~
w
w
w
w
w
w
w
w
w
w
w
• |Φ(.)|(Ω) ≤ −1
2
d
dt
Z
Ωu2(.) +
Z
Ωu(.) dµ(.) in D′(0, T )
• |Φ|(Q) ≤
Z Z
Qu dµ −
1
2
Z
Ωu(T )2 +
1
2
Z
Ωu2
0
• for a.e. t ∈ (0, T ), Φ(t) = ∇|Φ(t)|u(t), |Φ(t)| − a.e. in Ω
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 19 / 34
Main results
Notion of solution : Evolution Monge-Kantorovich equation
Theorem (Ig,2007)
For any u0 ∈ K and µ ∈ BV (0,T ,w∗-Mb(Ω)), the problem (EMK) has a weak solution (u,Φ) :
u ∈ C([0,T ); L2(Ω)) ∩ KT , ∂tu ∈ L
∞(0,T ; w
∗-Mb(Ω)), Φ ∈ L
∞(0,T ; w
∗-Mb(Ω)
n), u(0) = u0,
∂tu − ∇ · Φ = µ in D′(Q), and (u,Φ) satisfies
~
w
w
w
w
w
w
w
w
w
w
w
• |Φ(.)|(Ω) ≤ −1
2
d
dt
Z
Ω
u2(.) +
Z
Ω
u(.) dµ(.) in D′(0,T )
• |Φ|(Q) ≤
Z Z
Q
u σ dµ −1
2
Z
Ω
u(T )2 +1
2
Z
Ω
u20
• for a.e. t ∈ (0,T ), Φ(t) = ∇|Φ(t)|u(t), |Φ(t)| − a.e. in Ω
Moreover, u is the unique variational solution.
Theorem (Ig,2007)
For any µ ∈ L1(0,T ; w∗-Mb(Ω)) and u0 ∈ K , there exists (u,Φ) a weak solution of (EMK) i.e.
u ∈ C([0,T ); L2(Ω)) ∩ KT , u(0) = u0, Φ ∈ Mb(Q)n, ∂tu − ∇ · Φ = µ in D′(Q) and
supη∈C0(Ω)N , ‖η‖∞≤1
Z Z
Q
σΦ
|Φ|· η d|Φ| ≤
1
2
Z
T
0
Z
Ω
u2σt +
Z
T
0
Z
Ω
u σ dµ
for any σ ∈ D(0,T ). Moreover, u is the unique variational solution.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 20 / 34
Numerical approximation
Numerical approximation
Assume that µ = f ∈ L2(0, T ;L2(Ω)).
Time discretisation : Euler implicit schema.
0 < t0 < t1 < ... < tn = T whith ti − ti−1 = ε ∀i = 1, ..., n
f1, ..., fn ∈ L2(Ω)) such that
nX
i=1
Z ti
ti−1
‖f (t) − fi‖L2(Ω) ≤ ε.
consider the approximation of u(t) by :
uε(t) =
u0 if t ∈]0, t1]ui if t ∈]ti−1, ti ] i = 1, ...,n
where ui is given by :ui + ∂IIK (ui ) = εfi + ui−1
for i > 1 whith ui=0 = u0.
uε is the ε−approximate solution of (Pµ).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 21 / 34
Numerical approximation
Numerical approximation : convergence of the ε−approximate solution
Nonlinear semigroup theory (in L2(Ω))
⇓ ⇓ ⇓ ⇓ ⇓
Theorem
Let u0 ∈ L2(Ω) be such that |∇u0| ≤ 1 and f ∈ BV (0, T ;L2(Ω). Then,
1 For any ε > 0 and any ε−discretisation, there exists a unique ε−approximation uε of (Pf ).
2 There exists a unique u ∈ C([0,T ); L2(Ω)) such that u(0) = u0, and
‖u − uε‖C ([0,T );L2(Ω)) ≤ C(ε) → 0 as ε → 0.
3 The function u given by 2. is the unique solution of (Pµ).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 22 / 34
Numerical approximation
Numerical approximation : a generic problem
The generic problem is
v + ∂IIK (v) = g
⇓⇑ ⇓⇑ ⇓⇑ ⇓⇑
v = IPKg
where
K =n
z ∈ W 1,∞(Ω) ∩ W 1,20 (Ω) ; |∇u| ≤ 1
o
.
IPK the projection onto the convex K , with respect to the L2(Ω) norm :
v = IPK (u) ⇔ v ∈ K ,
Z
Ω(v − u)(v − z) > 0 for any z ∈ K .
Question :
For a given g ∈ L2, how to compute v = IPKg ?
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 23 / 34
Numerical approximation
Numerical approximation : Dual and Primal formulation
Standard duality argument (Ekeland-Temam) : if
P := minn
F (z) + H(Λz) ; z ∈ C10(Ω)
o
,
where
Λ : C0(Ω) −→ C(Ω)N
z −→ Λz := ∇z
F : C0(Ω) −→ R+
z −→ F (z) =1
2
Z
Ω|z − g |2
H : C(Ω)N −→ R
σ −→ H(σ) =
0 si |σ(x)| ≤ 1 ∀x ∈ Ω+∞ sinon ,
thenD := sup
p∈Y ′−
“
F∗(Λ∗p) + H∗(−p)”
≤ P.
If,
P < ∞
There exists u0 ∈ V such that F (u0) < ∞, H(Λu0) < ∞ and H is continuous at Λu0,
thenP = D.
andΛ∗p ∈ ∂F (u) et − p ∈ ∂H(Λu).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 24 / 34
Numerical approximation
Numerical approximation : Dual and Primal formulation
We have
Λ∗ :“
C(Ω)N”′
→“
C0(Ω)”′
Λ∗σ := ∇ · σ
F∗ :“
C0(Ω)”′
→ R+
F (z) =1
2
Z
Ω|z − g |2 +
1
2
Z
Ωg2
H∗ :“
C(Ω)N”′
→ R
H(σ) =
Z
Ω|σ|.
This implies that G(w) =1
2
Z
Ω( div(w))2 +
Z
Ωg div(w) +
Z
Ω|w | and
D = maxn
− G(σ) ; σ ∈ (C(Ω)N )∗o
.
So, one expects
−G(Φ) = maxn
− G(σ) ; σ ∈ (C(Ω)N )∗o
= minn
F (z) + H(Λz) ; z ∈ C10(Ω)
o
= F (u).
and−∇ · Φ = g − u and Φ ∈ ∂IIB(0,1)(∇u).
the natural space for the study of P by duality argument is the set of vector valued Radon measures suchthat the divergence is a L2.
This kind of argument was used by Prigozhin and Barrett.
Our aim is to simplify the analysis by using simply the space Hdiv (Ω) ;
Hdiv (Ω) =n
w ∈“
L2(Ω)”D
; div(w) ∈ L2(Ω)o
.Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 25 / 34
Numerical approximation
Numerical approximation : Dual and Primal formulation
J(z) =1
2
Z
Ω|z − g |2.
Hdiv (Ω) = σ ∈ (L2(Ω))N ; div(σ) ∈ L2(Ω)
G(σ) =1
2
Z
Ω(div(σ))2 +
Z
Ωgdiv(σ) + d
Z
Ω|σ|.
Theorem (Dumont-Ig,2007)
Let g ∈ L2(Ω) and v = IPk(g). Then, there exists a sequence (wǫ)ǫ>0 in Hdiv (Ω), such that, as
ǫ → 0,
limǫ→∞
“
− G(wǫ)”
= supw∈Hdiv (Ω)
“
− G(w)”
= minz∈L2(Ω)
J(z)
anddiv(wǫ) → v − g in L2(Ω).
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 26 / 34
Numerical approximation
Numerical approximation : main idea of the proof
supσ∈Hdiv (Ω)
“
− G(σ)”
≤ minz∈K
J(z).
We consider the following elliptic equation
(Sε)
8
>
>
>
<
>
>
>
:
vε −∇ · wε = g
wε = φǫ(∇vε)
˛
˛
˛
˛
˛
˛
in Ω
u = 0 on ∂Ω.
where, for any ε > 0, φε : RN → R
N is given by
φε(r) =1
ε(|r | − 1)+
r
|r |, for any r ∈ R
N .
⇓ ⇓ ⇓ ⇓ ⇓
For any g ∈ L2(Ω), (Sε) has a unique solution vε, in the sense that vε ∈ H10 (Ω),
wε := φε(∇vε) ∈ L2(Ω)N and vε −∇ · wε = g in D′(Ω).
⇓ letting ε → 0 ⇓
The proof of the theorem
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 27 / 34
Numerical approximation
Numerical approximation : approximation of infσ∈Hdiv (Ω)
G (σ)
We consider
V : Hdiv (Ω) ⊃ Vh : the space of finite elment of Raviard-Thomas
rh : Hdiv (Ω) → Vh the projection operator of Raviard-Thomas : for any w ∈ Hdiv (Ω), ash → 0, we have
(rh(w), div(rh(w)) → (w , div(w)) in“
L2(Ω)”N+1
Theorem (Dumont-Ig,2007)
Let qh be the solution ofG(qh) = infG(σh), σh ∈ Vh
and q ∈ Mb(Ω)N such that div(q) ∈ L2(Ω) and q is a solution of the solution of
G(q) = infG(σ), σ ∈ Hdiv (Ω),
then‖div(q) − div(qh)‖L2(Ω) → 0 as h −→ 0.
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 28 / 34
Films Cas d’une source centrale
Cas d’une source centrale
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 29 / 34
Films Ecoulement d’un tas contre un mur
Ecoulement d’un tas contre un mur
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 30 / 34
Films Ecoulement d’un tas sur une table
Ecoulement d’un tas sur une table
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 31 / 34
Films Cas d’une source qui tourne
Cas d’une source qui tourne
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 32 / 34
Films Effondrement d’un tas
Effondrement d’un tas
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 33 / 34
Related works
Related works and works in progress :
Nonlocale sandpile problem : compound Poisson process
Moving sand dunes : Barchanes
Sandpile problem on non flat regions (on obstacles) : lacs and rivers
Collapsing sandpile problem.
Hysterisis : angle of stability is different than the angle of avalanches.
Application to optimal mass transportation : how to use numerical results concerningsandpile to treat Monge problem ?Example : d(., ∂Ω)
Noureddine Igbida (LAMFA CNRS-UMR 6140, Universite de Picardie Jules Verne, 80000 Amiens)Sur l’equation de Monge-Kantorovich Le Havre, 16 Avril 2009 34 / 34