Joint works with Matt Jacobs and Wonjun Lee
Two mass densities over Ω⊂Rd, with same total mass,
∫Ωμ(x)dx=∫Ων(y)dy
μ
ν
How to optimally transport μ to ν?
Map T:Ω→Ω, measure μ
Pushforward: (T#μ)(A)=μ(T−1(A))
D E F I N I T I O N
μ
T#μ
T(x)
Tinf∫Ω∣T(x)−x∣2μ(dx)=:W22(μ,ν)
subject to: T#μ=ν
Cost to move mass μ(dx) from x to T(x) is
∣T(x)−x∣2μ(dx)
T(x)
Tinf∫Ω∣T(x)−x∣2μ(dx)=:W22(μ,ν)
subject to: T#μ=ν
Features:
1) A distance between μ and ν
2) An optimal map T
3) Formally a Riemannian metric on the manifold of measures with geodesics ρt=((1−t)x+tT(x))#μ
Numerically: not easy to solve
2048×2048 points
Caffarelli's counterexample
2048×2048 points
A distance W2 over measures, with “Riemannian” structure
“ρ˙t=−gradW2U(ρt)”
Functional U(ρ), want to make sense of
ρ(t+τ)=ρargminU(ρ)+2τ1W22(ρ(t),ρ)
Look at ”implicit Euler”
(JKO)
The right approach, theoretically and numerically
∂tρ+div(ρv)=0,
v=−∇ϕ,
ϕ=δU(ρ)
on domain Ω⊂Rd, no flux out and with initial condition ρ(t=0)=ρ0.
When τ→0,
Slow diffusion
U(ρ)=∫Ωm−11ρ(x)m+V(x)ρ(x)dx,
m>1. V(x) can be +∞.
Incompressible energy
U(ρ)=∫Ωu∞(ρ(x))+V(x)ρ(x)dx,
Aggregation-diffusion
U(ρ)=∫Ωρ(x)mdx+∬Ω∣x−y∣2ρ(x)ρ(y)dxdy
porous medium equation
∂tρ=Δρm.
m→1:U(ρ)=∫Ωρlogρ+Vρ
Primal, numerically difficult:
ϕ,ψsup⟨ψ,μ⟩−U∗(ϕ)
over (ϕ,ψ) s.t.
ψ(x)−ϕ(y)≤2τ∣x−y∣2.
ρinfU(ρ)+2τ1W22(μ,ρ).
Dual:
with ϕc(x)=infyϕ(y)+2τ∣x−y∣2,
ψc(y)=supxψ(x)−2τ∣x−y∣2.
☇
ϕsup⟨ϕc,μ⟩−U∗(ϕ)=:J(ϕ)
ψsup⟨ψ,μ⟩−U∗(ψc)=:I(ψ)
→ unconstrained concave maximization problems
→ Recover ρ∗ from ϕ∗ by
ρ∗=δU∗(ϕ∗)
ϕsup⟨ϕc,μ⟩−U∗(ϕ)=:J(ϕ)
ψsup⟨ψ,μ⟩−U∗(ψc)=:I(ψ)
U(ρ)=∫Ωu∞(ρ(x))+V(x)ρ(x)dx.
ρ(x)=(u∞∗)′(ϕ(x)−V(x)) guaranteed to be 0 on obstacle.
Remark
U∗(ϕ)=∫Ωu∞∗(ϕ(x)−V(x))dx
☇
H is the Sobolev space
∥h∥H2=∫Ωα1∣∇h(x)∣2+α2∣h(x)∣2dx
J(ϕ)=⟨ϕc,μ⟩−U∗(ϕ)
I(ψ)=⟨ψ,μ⟩−U∗(ψc)
Recall
Jacobs, Lee, L. (’21)
A L G O R I T H M
Assume that
0≤−δ2J(ϕ)(h,h)≤ ∥h∥H2
for any ϕ,h∈H. Then the iterations
ϕk+1=ϕk+∇HJ(ϕk)
converge to the supremum of J.
Fundamental lemma in optimization:
Want Hessian bound
0≤−δ2J(ϕ)(h,h)≤∥h∥H2
J=F−U∗ with F(ϕ)=∫Ωϕcdμ.
−δ2F(ϕ)(h,h)=τ∫Ω∣∇h(x)∣g(ϕ)2(Tϕ#μ)(dx)
U∗(ϕ)=∫Ωu∞∗(ϕ(x))dx
δ2U∗(ϕ)(h,h)≤Ctrace∫Ω~∣∇h(x)∣2+∣h(x)∣2dx
δ2U∗(ϕ)(h,h)=∫{ϕ=0}∣h(z)∣2dσ(z)
Operations on a grid with N points:
- c-transform: O(N)
- Tϕn:O(N)
- Δ−1:O(NlnN)
Novelty of the approach:
1) a careful analysis of the Hessian of the objective function
2) the back-and-forth scheme which boosts the convergence
Slow diffusion (porous medium eq)
V(x)=−sin(5πx1)sin(3πx2)
512×512 points
m=2
m=4
Slow diffusion
m=4
V(x)=∥x−a∥2
512×512 points
Incompressible
V(x)=∥x−a∥2
1024×1024 points
Aggregation-diffusion
U(ρ)=∫ρ(x)3dx+∬∣x−y∣2ρ(x)ρ(y)dxdy