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Arnold interpretation Approached geodesics scheme Optimal Transport Semi-Discrete OT Numerical Scheme Bonus

A Lagrangian scheme ”a la Brenier” for theincompressible Euler equations

T. O. Gallouet1 , Q. Merigot 2,E. Shwindt 3, B. Levy 3,

F.X. Vialard 1

Inria, Paris 1

Universite Paris Sud-Orsay 2

Inria, Nancy 3

Paris 6, LJLL, 16 march 2018


Incompressible Euler equations

Domain: Ω ⊆ Rd with Leb measure. Eulerian formulation:∂tu(t, x) + (u(t, x) · ∇) u(t, x) = −∇p(t, x),

div(u(t, x)) = 0

u(t, x) · n = 0

u(0, x) = u0.

Lagrangian formulation:ddtφ(t, x) = u (t, φ(t, x)) for t ∈ [0,T ], x ∈ Ω ,

φ(0, ·) = id,

∂tφ(0, ·) = u0.


Incompressible Euler equations II

Measure preserving maps:

S =s ∈ L2(Ω,Rd) | s# Leb = Leb

,

The incompressibility constraint reads

div u = 0 ⇐⇒ det∇φ(t) = 1 ⇐⇒ φ(t, ·) ∈ S

The evolution equation

d2

dt2φ(t) = −∇p(t, φ(t, x)) ∈ (TφS)⊥

Conclusion: solutions of the incompressible Euler equations aregeodesics of the measure preserving maps S for the L2 metric.


Approached geodesics a la Brenier

Simple example: we consider R× 0 ⊂ R2.

geodesic: γ: [0,T ]→ R2 withγ(t) = (t, 0), t ∈ [0,T ],

γ(0) = (0, 0),

γ(0) = (1, 0).

Approached geodesic z with initial error δ.z(0) = (0, δ),

z(0) = (1, 0).

ε-evolution :

z(t) =1

ε2(PR(z)− z)


Approached geodesics a la Brenier II

Exact solution:

z(t) =(t, δ cos

t

ε

).

Solution for the Hamiltonian system defined with

H(z , v) =1

2||v ||2 +

1

2ε2d2R×0(z)

Quantifying the convergence: the modulated energy

Eγ(t) =1

2||z(t)− γ(t)||2 +

1

2ε2d2R×0(z(t)).

Eγ(t) =δ2

ε2


naive idea

Compute the solution of the Hamiltonian system associated to

H(f , v) =1

2‖v‖2

M +d2S (f )

2ε2.

How to compute d2S (f ) ?

→ Optimal Transport (Brenier)


Settings

• two probability measures µ, ν on Rd .

• cost c : X × Y → R, in particular c(x , y) = |x − y |2.

• t#µ = ν: for any measurable set B we haveν(B) = µ

(t−1(B)

).

Problem (Monge 1781)

Find T such that

T = argmint#µ=ν

(∫c (x , t(x)) dµ(x)

).

MK22(µ, ν) = min

t#µ=ν

(∫|t(x)− x |2 dµ(x)

)


Existence

Theorem (Existence, Brenier)

Let µ ac. w.r.t. Leb. Then there exists a unique ∇ϕ with ϕconvex such that T = ∇ϕ.

Theorem (Polar decomposition, Brenier)

Let f ∈ L2(Ω,Rd) and ν = f# Leb. Then there exists ϕ convexand σ ∈ S such that

• f = ∇ϕ σ,

• d2S (f ) = MK2

2(Leb, ν) = MK22(Leb, f# Leb),

• σ ∈ PS(f ).


Kantorovich potentials

Owner point of view:

Factory points: Xα, α ∈ 1, ...,N, producing 1N .

Distribution points: Yβ, β ∈ 1, ...,N, distributing 1N .

Transport cost: c(Xα,Yβ).

Shipper point of view: At a factory Xα, I buy −ϕ(Xα). At a

distribution point Yβ, I sell ψ(Yβ). I must ship everything and I

guarantee for all α, β

(∗∗) ϕ(Xα) + ψ(Yβ) ≤ c(Xα,Yβ).


Kantorovich potentials II

The (∗∗) deal means∫ϕ(x)dµ(x) +

∫ψ(y)dν(y) ≤ MK2

2(µ, ν).

Kantorovich duality

supϕ⊕ψ≤c

∫ϕ(x)dµ(x) +

∫ψ(y)dν(y) = MK2

2(µ, ν).

c-transform

ψc(x) = supy

(c(x , y)− φ(y)) .


Semi-Discrete optimal transport (Q. Merigot)

We want to compute MK22

(Leb, 1

N

∑Nβ=1 δYβ

).

Laguerre Cells for points Yβ and prices ψβ.

Lagα ((Yβ)1...N , (ψβ)1...N) =x ∈ Ω

∣∣∣−ψα + |x − Yα|2 ≤ −ψβ + |x − Yβ|2 , for all β

→ Partition of the space Ω.

To be sure that the price gives a transport maps, it must holds

|Lagα (Yβ, ψβ)| = 1/N.

Numerically: Newton on this condition.


Discretization space

Ω partitioned in N cells Pα with mass 1N .

MN = piecewise constant functions on Pα ⊂ L2(Ω,Rd).

For f ∈MN :

d2S (f ) = MK2

2(Leb, f# Leb) = MK22(Leb,

1

N

N∑α=1

δfα).

Projection set of f on S:

PS(f ) = σ ∈ S | ∀α, σ(Pα) = Lagα(fβ, ψopt).


Approximate Geodesics

d2S as a function : MN → R:

H(f , v) =1

2‖v‖2

M +d2S (f )

2ε2

=1

2‖v‖2

M +1

2ε2MK 2

2 (Leb,1

N

N∑α=1

δfα).

Scheme: f (t) +∇d2

S (f (t))

2ε2 = 0, for t ∈ [0,T ] ,

(f (0), f (0)) ∈M2N

With1

2∇d2

S (f (t)) = f − PM PS(f ) ∈MN


Approximate Geodesics II

PM PS(f ) is the application

Pα → Bar(Lagα(fβ, ψ

opt))

Scheme rewritesf (t) = PMPS(f )−fε2 = 0, for t ∈ [0,T ] ,

(f (0), f (0)) ∈M2N

Full discretization (Euler semi-symplectic), step time τ .V n+1 = V n − τ f n−PMN

PS(f n)

ε2

f n+1 = f n + τV n+1,


ConvergenceLet u be a smooth (eulerian) solution of the incompressible Eulerequations.

The modulated energy:

Eu(t) =1

2‖f (t)− u(t, f (t))‖2

M +d2S (f )

2ε2.

Gronwall estimates, Semi-Discrete:

Eu(t) ≤ C

[ε2 + hN +

h2N

ε2

]Full discrete:

Eu(t) ≤ C

[ε2 + hN +

h2N

ε2+ κ+

τ

ε

]Where

κ = maxn∈N∩[0,T/τ ]

(Hn − H0

)≤ τ

ε2.


The Riemannian submersion

Space: (S,L2) ⊂ L2(Ω)

Projection: d2S (f (t)) = MK2

2(Leb, f# Leb), polar decomposition.

Geodesics: Incompressible Euler equations.







Behind the structure: the Otto Riemannian submersion

π : (Diff(Ω),L2)→ (Dens(Ω), MK2)

φ 7→ φ#(Leb).

DefinitionThe map π is a Riemannian submersion if π is a submersion andfor any x ∈ M, the map dfx : Ker(dπx)⊥ 7→ Tπ(x)N is an isometry.









φ 7→ φ#(Leb).


S = π−1(Leb), ∂tρ = div(ρv), v = φ(t) φ−1(t)


The Riemannian submersionSpace: (S,L2) ⊂ L2(Ω)






φ 7→ φ#(Leb).


S = π−1(Leb), ∂tρ = div(ρv), v = φ(t) φ−1(t)

v = u+∇p, div(u) = 0, W22(∂tρ) = inf

∂tρ=div(ρv)‖u‖L2(ρ) = ‖∇p‖L2(ρ)



What if

Projection: given by the WMKHFR distance on M+(Ω)



What if

Projection: given by the HK distance on M+(Ω)



What if

Projection: given by the d distance on M+(Ω)



What if

Projection: given by the WFR distance on M+(Ω)



What if

Projection: given by the HK-WFR distance on M+(Ω)



Yes





Submersion

π :(Diff(Ω) n C∞(Ω,R+), dc

)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).




Submersion


)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).

S = π−1(Leb) =(φ,√

Jac(φ))

, ∂tρ = div(ρv) + ρr ,




Submersion


)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).

S = π−1(Leb) =(φ,√

Jac(φ))


WFR22(∂tρ) = inf

∂tρ=div(ρv)+rρ‖v‖2

L2(ρ) + ‖r‖2L2(ρ) = gc(∇r , r)



Space: S ⊂ Diff(Ω) n C∞(Ω,R+)


Submersion


)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).

S = π−1(Leb) =(φ,√

Jac(φ))


WFR22(∂tρ) = inf


L2(ρ) + ‖r‖2L2(ρ) = gc(∇r , r)


The Riemannian submersionSpace: S ⊂ Diff(Ω) n C∞(Ω,R+)

Projection: given by the HK-WFR distance on M+(Ω) New polardecomposition

Geodesics:

Submersion


)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).

S = π−1(Leb) =(φ,√

Jac(φ))


WFR22(∂tρ) = inf


L2(ρ) + ‖r‖2L2(ρ) = gc(∇r , r)


The Riemannian submersionSpace: S ⊂ Diff(Ω) n C∞(Ω,R+)

Projection: given by the HK-WFR distance on M+(Ω) New polardecomposition

Geodesics: Camassa-Holm equations.

Submersion


)→ (M+(Ω), WFR)

(φ, λ) 7→ π(φ, λ) = φ#(λ2 Leb).

S = π−1(Leb) =(φ,√

Jac(φ))


WFR22(∂tρ) = inf


L2(ρ) + ‖r‖2L2(ρ) = gc(∇r , r)


ConclusionRevisit Brenier’s work for the incompressible Euler equations in thiscase

• Generalized solutions

• Regularity of the pression

• Numerical scheme

New polar decompostion:

(φ, λ) = expC(M)

(−1

2∇pz0 ,−pz0

) (s,

√Jac(s))

or equivalently

(φ, λ) =

(ϕ, e−z0

√1 + ‖∇z0‖2

)· (s,

√Jac(s)) ,

where pz0 = ez0 − 1 and

ϕ(x) = expMx

(− arctan

(1

2‖∇z0(x)‖

)∇z0(x)

‖∇z0(x)‖

).

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