Optimal Transport versus Growth and Decay · Chizat-Peyré-Schmitzer-Vialard3...

Optimal Transport versusGrowth and Decay

Alexander Mielke

Weierstraß-Institut für Angewandte Analysis und Stochastik, BerlinInstitut für Mathematik, Humboldt-Universität zu Berlin

www.wias-berlin.de/people/mielke/

Calculus of Variations and Optimal TransportationInternational Conference in Honor of Yann Brenier

Paris, Institut Henri Poincaré10 – 13 janvier 2017

Main task: Find distances on measure of possibly unequal mass• model transport via Monge-Kantorovich-Wasserstein distance• model growth and decay “similarly”

Fisher-Rao distance

• final distance = inf-convolution of these two measures

Co-workers on theHellinger-Kantorovich distance1

Matthias Liero (WIAS Berlin)Giuseppe Savaré (Pavia)

Parallel work using the name Wasserstein-Fisher-Rao distanceKondratyev-Monsaingeon-Vorotnikov2

Chizat-Peyré-Schmitzer-Vialard3

Fisher-Rao = Hellinger distance = Hell(µ0, µ1)2 =

(√dµ0dµ−√

dµ1dµ

)2dµ.

arXiv papers: 1 1508.07941, 1509.00068 2 1505.07746 3 1506.06430, 1508.05216

A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 2 (29)

Main task: Find distances on measure of possibly unequal mass• model transport via Monge-Kantorovich-Wasserstein distance• model growth and decay “similarly” Hellinger-Kakutani distance• final distance = inf-convolution of these two measures

(√dµ0dµ−√

dµ1dµ

)2dµ.

arXiv papers: 1 1508.07941, 1509.00068

2 1505.07746 3 1506.06430, 1508.05216

Main task: Find distances on measure of possibly unequal mass• model transport via Monge-Kantorovich-Wasserstein distance• model growth and decay “similarly” Fisher-Rao distance• final distance = inf-convolution of these two measures

(√dµ0dµ−√

dµ1dµ

)2dµ.

arXiv papers: 1 1508.07941, 1509.00068 2 1505.07746 3 1506.06430, 1508.05216

Overview

1. Motivation

2. Geodesic curves and dissipation distances

3. Characterizations of HKα,β

4. Geodesic λ-convexity for HK

5. A fast-diffusion equation with growth and decay

A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017

1. Motivation

Imaging:interpolation between images as geodesics

• amounts of colors may be different• even if the amount of green is the same, you may wantto grow the tree and shrink the grass

1. Motivation

Cloud motion shows transport, growth, and decay

Rain in France yesterday evening

1. Motivation

My personal motivation: semiconductor physicsn(t, x) density of free electronsn = α∆n︸︷︷︸

=diffusion

+ div(n∇V (x)

)︸︷︷︸=drift by electric field

+ β(neq(x)− n)︸︷︷︸=emission/absorption

reservoir of absorbed electrons(bound in acceptor traps)freely moving electrons n(t, x)

If neq(x) = n0e−V (x), this PDE has the gradient-flow form

n = −K(n)DE(n) =: −∇KE(n)

rel. entropy E(n) :=∫

ΩλB(n/neq)neq dx with λB(z) = z log z − z + 1.

metric tensor K(n)ξ := − div(αn∇ξ

)︸︷︷︸transport part

+ βn−neq

log(n/neq)ξ︸︷︷︸growth/decay part

K(n) = K(n)∗ ≥ 0︸︷︷︸Onsager operator

1. Motivation

My personal motivation: semiconductor physicsn(t, x) density of free electronsn = α∆n︸︷︷︸

=diffusion

+ div(n∇V (x)

)︸︷︷︸=drift by electric field

+ β(neq(x)− n)︸︷︷︸=emission/absorption

reservoir of absorbed electrons(bound in acceptor traps)freely moving electrons n(t, x)

If neq(x) = n0e−V (x), this PDE has the gradient-flow form

n = −K(n)DE(n) =: −∇KE(n)

rel. entropy E(n) :=∫

ΩλB(n/neq)neq dx with λB(z) = z log z − z + 1.

metric tensor K(n)ξ := − div(αn∇ξ

)︸︷︷︸transport part

+ βn−neq

log(n/neq)ξ︸︷︷︸growth/decay part

K(n) = K(n)∗ ≥ 0︸︷︷︸Onsager operator

1. Motivation

Formal gradient structure also works for Reaction-Diffusion Systemsif the detailed-balance condition holds (microscopic reversibility)!Reaction corresponds to growth and decay of individual speciesvan Roosbroeck system for semiconductors− div(ε∇φ) = d−n+p electrostatics

n = div(µn(∇n−n∇φ)

)+ g − rnp electron balance

p = div(µp (∇p+ p∇φ)

)+ g − rnp hole balance

g = generation rate of electron/hole pairs, r = recombination rate

Gradient structure (n, p) = −KvR(n, p)DEvR(n, p) [M.2011 Nonlinearity]

Energy EvR(n, p) =∫

ΩλB(n/w)w+λB(p/w)w+ ε

2 |∇φn,p|2 dx with w =

√g/r

Onsager operator KvR(n, p)(ξnξp

)= −

(div(nµn∇ξn)

div(pµp∇ξp)

)︸︷︷︸

transport

+ gΛ(1, npw2

)(1 11 1

)(ξnξp

)︸︷︷︸reaction

with Λ(a, b) = a−blog a/b

1. Motivation

Formal gradient structure also works for Reaction-Diffusion Systemsif the detailed-balance condition holds (microscopic reversibility)!Reaction corresponds to growth and decay of individual speciesvan Roosbroeck system for semiconductors− div(ε∇φ) = d−n+p electrostatics

n = div(µn(∇n−n∇φ)

)+ g − rnp electron balance

p = div(µp (∇p+ p∇φ)

)+ g − rnp hole balance

g = generation rate of electron/hole pairs, r = recombination rate

Gradient structure (n, p) = −KvR(n, p)DEvR(n, p) [M.2011 Nonlinearity]

Energy EvR(n, p) =∫

ΩλB(n/w)w+λB(p/w)w+ ε

2 |∇φn,p|2 dx with w =

√g/r

Onsager operator KvR(n, p)(ξnξp

)= −

(div(nµn∇ξn)

div(pµp∇ξp)

)︸︷︷︸

transport

+ gΛ(1, npw2

)(1 11 1

)(ξnξp

)︸︷︷︸reaction

with Λ(a, b) = a−blog a/b

Overview

1. Motivation

Given a gradient system (Q,E,K) we may try to identify the associatedgeometry:dissipation distance dK for metric tensor G = K−1

constant-speed geodesic curves γ : [0, 1]→ Q are minimizers for

dK(q0, q1)2 := inf ∫ 1

0〈G(γ)γ, γ〉 ds

∣∣ γ ∈ C1([0, 1];Q), γ(0) = q0, γ(1) = q1

A functional E : Q→ R∞ is called geodesically λ-convex, if[0, 1] 3 s 7→ E(γ(s))− λ

2 s2dK(γ(0), γ(1))2 is convex

for all such geodesic curves.

E(γ(s))

Given a gradient system (Q,E,K) we may try to identify the associatedgeometry:dissipation distance dK for metric tensor G = K−1

constant-speed geodesic curves γ : [0, 1]→ Q are minimizers for

dK(q0, q1)2 := inf ∫ 1

∣∣ γ ∈ C1([0, 1];Q), γ(0) = q0, γ(1) = q1

A functional E : Q→ R∞ is called geodesically λ-convex, if

[0, 1] 3 s 7→ E(γ(s))− λ2 s

2dK(γ(0), γ(1))2 is convexfor all such geodesic curves.

E(γ(s))

Observations: in Ambrosio-Gigli-Savaré 2005(Otto-Westdickenberg 2005, Daneri-Savaré 2008)

Solutions q : [0,∞[→ Q ofgeodesically λ-convex gradient systems (Q,E,K) satisfy

(EVI) ∀ w ∈ Q : 12

ddtdK(q(t), w)2 + λ

2 dK(q(t), w)2 ≤ E(w)− E(q(t))

Hence, they enjoy nice properties for metric gradient flows (Q,E, dK):

• dK(q1(t), q2(t))) ≤ e−λtdK(q1(0), q2(0)) for t > 0 contraction/Lipschitz

• dK(q(t+τ), q(t))) ≤ 2λ (1−e−λτ )

(E(q(t))− inf E

)uniform continuity

• E(q(t)) ≤ λ2(eλt−1)

dK(q(0), w)2 + E(w) for all w regularization

• time discretization qτk+1 = argminE(q)+ 12τ dK(qτk , q)

=⇒ dK(q(t), qτ (t)) ≤ C(q0)√τ e−λτ t convergence

How can we characterize dissipation distances defined via K(q)?

dK(q0, q1)2 := inf ∫ 1

∣∣ γ ∈ C1([0, 1];Q), γ(0) = q0, γ(1) = q1

For pure diffusion equations this was successful (JKO, Brenier,...AGS2005):

KMKW(c)ξ = − div(c∇ξ) for ξ ∈ H1(Ω) gives

Monge-Kantorovich-Wasserstein distance ←→ optimal transport

dMKW(µ0, µ1)2 := inf∫

Ω×Ω|x−y|2η(dx, dy)

∣∣∣ η ∈ C(µ0, µ1)

Geodesics of transport type γMKW(s) =((1−s)id + sT )#γ(0)

Pure growth and decay (Hellinger = Fisher-Rao)KHell(c)ξ = 4cξ ⇐⇒ G(c)c = 1

4c c if µ = c dx

Hell(c0dx, c1dx)2 = inf ∫ 1

∫Ωc2

4c dx dt∣∣ c0 c(t,·)

(√c1 −

√c0)2

For general measures in M(Rd) the geodesic curves are given by

γHell(s) =((1−s)

√θ0 + s

√θ1

)2(µ0+µ1), where θj =

dµjd(µ0+µ1)

Other forms for Kreact would be possible like Kreact(c)ξ) = κ(c)︸︷︷︸≥0

The Hellinger distance is distinguished: K linear in c !

4c c if µ = c dx

∫Ωc2

(√c1 −

√c0)2

γHell(s) =((1−s)

√θ0 + s

√θ1

dµjd(µ0+µ1)

4c c if µ = c dx

∫Ωc2

(√c1 −

√c0)2

γHell(s) =((1−s)

√θ0 + s

√θ1

dµjd(µ0+µ1)

General RDS c = div(M(c)∇c)−R(c) = −KRDS(c)DE(c)

with Onsager operator KRDS(c)ξ = − div(M(c)∇ξ) + Kreact(c)ξ

dK(q0, q1)2 = inf∫ 1

0〈G(q(s)q(s), q(s)〉 ds | q0

q(s) q1 but G implicit

Benamou-Brenier (2000) trick (use “G(q) = K(q)−1” with K explicit):

dK(q0, q1)2 = inf∫ 1

0〈ξ(s),K(q(s))ξ(s)〉 ds | q = K(q)ξ, q(0)=q0, q(1)=q1

For the Onsager operator KRDS we obtain amodified continuity equation (mCE) with source terms from reactions:dKRDS(c0, c1)2 =

inf∫ 1

Ξ:M(c):Ξ + ξ·Kreact(c)·ξ dx ds | (mCE), c(0)=c0, c(1)=c1

(mCE) c+ div(M(c)Ξ

)−Kreact(c)ξ = 0

Wellposedness results (i.e. inf = min > 0) by Matthes et al. 2015,16,...

dK(q0, q1)2 = inf∫ 1

0〈G(q(s)q(s), q(s)〉 ds | q0

dK(q0, q1)2 = inf∫ 1

inf∫ 1

(mCE) c+ div(M(c)Ξ

)−Kreact(c)ξ = 0

dK(q0, q1)2 = inf∫ 1

0〈G(q(s)q(s), q(s)〉 ds | q0

dK(q0, q1)2 = inf∫ 1

inf∫ 1

(mCE) c+ div(M(c)Ξ

)−Kreact(c)ξ = 0

Only one case is fully understood, namely the simplestOnsager operator for a scalar density c ≥ 0:

Kα,β(c)ξ = − div(αc∇ξ) + βc ξ (diffusion + absorption/generation)

HKα,β(c0,c1)2 = inf

(α|∇ξ|2+βξ2

)c dxdt

∣∣∣∣mCE, c(0) = c0, c(1) = c1

(mCE) c+ α div

(c∇ξ)− βcξ = 0

Independently studied by

• Kondratyev-Monsaingeon-Vorotnikov arXiv:1505.07746• Chizat-Peyré-Schmitzer-Vialard arXiv:1506.06430 and arXiv:1508.05216• Liero-M.-Savaré arXiv:1508.07941=SIMA’16 and arXiv:1509.00068

(α|∇ξ|2+βξ2

)c dxdt

∣∣∣∣mCE, c(0) = c0, c(1) = c1

(mCE) c+ α div

(c∇ξ)− βcξ = 0

For β > 0 there is no mass conservation, henceHKα,β is a distance metric on all of M(Ω) (all non-negative measures)

α > 0 and β = 0:Only diffusion, no reaction: HKα,0(µ0, µ1)2 = 1

αdMKW(µ0, µ1)2

(and =∞ if µ0(Ω) 6= µ1(Ω) )

α = 0 and β > 0No diffusion, only reaction: HK0,β(µ0, µ1)2 = 4

βHell(µ0, µ1)2

(√µ0 −

√µ1

(α|∇ξ|2+βξ2

)c dxdt

∣∣∣∣mCE, c(0) = c0, c(1) = c1

(mCE) c+ α div

(c∇ξ)− βcξ = 0

For β > 0 there is no mass conservation, henceHKα,β is a distance metric on all of M(Ω) (all non-negative measures)

α > 0 and β = 0:Only diffusion, no reaction: HKα,0(µ0, µ1)2 = 1

αdMKW(µ0, µ1)2

(and =∞ if µ0(Ω) 6= µ1(Ω) )α = 0 and β > 0No diffusion, only reaction: HK0,β(µ0, µ1)2 = 4

βHell(µ0, µ1)2

(√µ0 −

√µ1

2. Geodesic curves and dissipation distancesinitial measureµ1 = χ[1,2]

final measureµ2 = δ0

left: α = 0Hellinger geodesics((1−s)√µ0+s

√µ1

middle: β = 0Wassersteintransport

right: αβ > 0Hellinger-Kantorovichgeodesic(transport andabsorp./gener.)

Hellinger-Kakutani Wasserstein-Kantorovich Hellinger-Kantorovich

√µ1

)2middle: β = 0Wassersteintransport

√µ1

)2middle: β = 0Wassersteintransport

For α = 1 and β = 4 move µ0 =30∑k=0

δ(1,k/15) to µ1 = 31 δ(0,0)

Overview

1. Motivation

Equation for geodesic curves c : [0, 1]×Ω→ ]0,∞[

(mCE) c = Kα,β(c)ξ = −α div(c∇ξ) + βξc

(gHJ) ξ + α2 |∇ξ|

2 + β2 ξ

The linear dependence of Kα,β(c) on c makes (gHJ) independent of c!!

For notational simplicity we mostly choose α = 1, β = 4 and set HK = HK1,4.

Characterization of HK via Hamilton–Jacobi equationHK(µ0, µ1)2 = sup

ξ:[0,1]×Ω→R2( ∫

Ωξ(1, x) dµ1 −

∫Ωξ(0, x) dµ0

)for all ξ satisfying ∂tξ + 1

2 |∇ξ|2 + 2ξ2 ≤ 0.

Is there a generalization of the well-known Oleinik-Lax-Hopf formula

for ∂tξ + 12 |∇ξ|

2 ≤ 0, namely ξ(t, x) = infy∈Rd(|x−y|2

2t + ξ(0, y))?

Equation for geodesic curves c : [0, 1]×Ω→ ]0,∞[

(mCE) c = Kα,β(c)ξ = −α div(c∇ξ) + βξc

(gHJ) ξ + α2 |∇ξ|

2 + β2 ξ

The linear dependence of Kα,β(c) on c makes (gHJ) independent of c!!

For notational simplicity we mostly choose α = 1, β = 4 and set HK = HK1,4.

Characterization of HK via Hamilton–Jacobi equationHK(µ0, µ1)2 = sup

ξ:[0,1]×Ω→R2( ∫

Ωξ(1, x) dµ1 −

∫Ωξ(0, x) dµ0

)for all ξ satisfying ∂tξ + 1

2 |∇ξ|2 + 2ξ2 ≤ 0.

Is there a generalization of the well-known Oleinik-Lax-Hopf formula

for ∂tξ + 12 |∇ξ|

2 ≤ 0, namely ξ(t, x) = infy∈Rd(|x−y|2

2t + ξ(0, y))?

HK(µ0, µ1)2 = sup 2

(〈ξ(1), µ1〉−〈ξ(0), µ0〉) subj. to ∂tξ + 1

2|∇ξ|2 + 2ξ2 ≤ 0

Explicit solution formula for solutions of (gHJ)

ξ(t, x) = infx0∈Ω

Pt(|x−x0|, ξ(0, x0)) with Pt(r, ξ) = 12t

(1− cos2(minr,π/2)

1+2tξ

)Characterization of HK via Kantorovich potentialsHK(µ0, µ1)2 = sup

ϕ0,ϕ1

( ∫Ω

(1−e−ϕ0) dµ0 +∫

Ω(1−e−ϕ1) dµ1

)subject to the constraint ϕ0(x0)+ϕ1(x1) ≤ c(x0, x1)

with cost function c(x, y) =

−2 log(cos |x−y|) for |x−y| < π/2,

∞ otherwise.

Here ϕ1(x) = − log(1−2ξ(1, x)) and ϕ0(x) = − log(1+2ξ(0, x))

Moving Dirac mass µ(s) = a(s)δx(s) with a(s) ≥ 0 and x(s) ⊂ Ω ⊂ Rd

... is geodesic curve iff (a, x) minimizes∫ 1

a(s)+a(s)

α|x(s)|2

... is given exactly (w.l.o.g. α = 1, β = 4 from now on) bya(s) = (1−s)2a0 + s2a1 + 2s(1−s)√a0a1 cosL with L = |x1−x0|π

x(s) = (1−ρ(s))x0+ρ(s)x1 with ρ(s) = 1L arctan

(s√a1/a0 sinL

1−s+s√a1/a0 cosL

)... gives the conjecture

HK(a0δx0, a1δx1

a0+a1 − 2

√a0a1 cos |x1−x0| for |x1−x0| ≤ π,

a0 + a1 + 2√a0a1 for |x1−x0| ≥ π.

However, for π/2 < |x1−x0| < π the “non-transport”/Hellinger curveµ(s) = (1−s)2δx0︸︷︷︸

annihilation

+ s2δx1︸︷︷︸creation

gives length a0 + a1 < a0 + a1 + 2√a0a1

a(s)+a(s)

α|x(s)|2

(s√a1/a0 sinL

HK(a0δx0, a1δx1

a0+a1 − 2

√a0a1 cos |x1−x0| for |x1−x0| ≤ π,

a0 + a1 + 2√a0a1 for |x1−x0| ≥ π.

annihilation

a(s)+a(s)

α|x(s)|2

(s√a1/a0 sinL

HK(a0δx0, a1δx1

a0+a1 − 2

√a0a1 cos |x1−x0| for |x1−x0| ≤ π,

a0 + a1 + 2√a0a1 for |x1−x0| ≥ π.

annihilation

HK(a0δx0 , a1δx1 )2 = a0 + a1 − 2√a0a1 cos |x1−x0| for |x1−x0| ≤ π/2

Idea: set aj = r2j and

define the cone CΩ = ([0,∞[×Ω)/∼

with distance

dCΩ((r0, x0), (r1, x1))2

:= r20 +r2

1−2r0r1 cos(min|x1−x0|, π

Characterization of HK via lifts to cone CΩ (CPSVialard’15)

HK(µ0, µ1)2 = inf ∫

CΩ×CΩ

dCΩ(z0, z1)2 dγ(z0, z1)

∣∣∣ γ ∈ Ccone(µ0, µ1)

where Ccone(µ0, µ1) :=γ ∈M(CΩ × CΩ)

∣∣ ∫[0,∞[

r2Πj#γ = µj

Characterization via lifts to cone CΩ

HK(µ0, µ1)2 = inf ∫

CΩ×CΩ

dCΩ(z0, z1)2 dγ(z0, z1)

∣∣∣ γ ∈ Ccone(µ0, µ1)

where Ccone(µ0, µ1) :=γ ∈M(CΩ×CΩ)

∣∣ ∫[0,∞[

r2Πj#γ dr = µj

Corollary: Geodesic curves (Liero-M-Savaré SIMA’16)If γ ∈ Ccone(µ0, µ1) is an optimal transport plan for HK(µ0, µ1)2, then

the curve s 7→ µ(s) =

∫ ∞r=0

r2Π(1−s,s)# γ dr ∈M(Ω)

is a geodesic curve in (M(Ω),HK). Moreover, all geodesics in (M(Ω),HK)are such projections of Wasserstein geodesics on (M(CΩ),WassdCΩ

(M(Ω),HK) is a complete geodesic space andHK-convergence is equivalent to weak convergence

Characterization of HK via entropy regularization

HK(µ0, µ1)2 = infη

(∫Ω

λB( dΠ0

)dµ0 +

λB( dΠ1

)dµ1 +

∫Ω×Ω

c(x0, x1) dη

)with transport plans η ∈ CΩ(µ0, µ1) :=

η ∈M(Ω×Ω)

∣∣Πj#η µj

This very useful form can be obtained in two independent ways:

by dualization of the characterization via the Kantorovich potentials

by the following elementrary identity and the cone construction

r20 + r2

1 − 2r0r1 cos(minr, π2 ) = minϑ>0

(r20λB

)+ λB

)+ ϑc(r)

Boltzmann’s entropy function λB(z) = z log z−z+1 (i.e.λ′B(z) = log z)

and the cost function c(r) =

)for r < π/2,

∞ otherwise.

Overview

1. Motivation

Question:Which energies E(c) =

∫ΩE(c(x)) dx are geodesically λ-convex for Kα,β?

There is a rich theory for SCALAR drift-diffusion equations

∂tc = div(µ(x, c)∇c+ a(x, c)∇V (x)

), t > 0, x ∈ Ω

using optimal transport and Wasserstein metric on probability spacesOtto’98,... , Ambrosio&Gigli & Savaré’05, ...Otto-Westdickenberg’05, Villani’09, ... (for K1,0 = KWass)

Optimal transport needs a continuous underlying space with geodesic curves.

However, this theory does not apply tosystems of equationsequations with reaction terms

Question:Which energies E(c) =

∫ΩE(c(x)) dx are geodesically λ-convex for Kα,β?

There is a rich theory for SCALAR drift-diffusion equations

∂tc = div(µ(x, c)∇c+ a(x, c)∇V (x)

), t > 0, x ∈ Ω

using optimal transport and Wasserstein metric on probability spacesOtto’98,... , Ambrosio&Gigli & Savaré’05, ...Otto-Westdickenberg’05, Villani’09, ... (for K1,0 = KWass)

Optimal transport needs a continuous underlying space with geodesic curves.

However, this theory does not apply tosystems of equationsequations with reaction terms

Daneri-Savaré’08 (building on Otto-Westdickenberg’05)provide a differential characterization of GλC.This is applicable to general gradient flows (systems, nonconservative)!

Under suitable conditions the gradient-flow equation (PDE)q = −K(q)DE(q) = −V (q)

generates a nice semiflow q(t) = S(t, q(0)) on a dense set of nice functions.Then, it is justified to calculate the (co-variant) Hessian H = HessK(E) as aquadratic form as follows

H(q, ξ) := 〈ξ,K(q)DV (q)∗ξ〉 − 12Dq〈ξ,K(q)ξ〉[V (q)]

Theorem [Liero-M.’13]: Under suitable conditions E is GλC for dK if andonly if H(q, ξ) ≥ λ〈ξ,K(q)ξ〉 on the dense sets for q and ξ.

Assumptions difficult, but necessary conditions follows easily.

Daneri-Savaré’08 (building on Otto-Westdickenberg’05)provide a differential characterization of GλC.This is applicable to general gradient flows (systems, nonconservative)!

Under suitable conditions the gradient-flow equation (PDE)q = −K(q)DE(q) = −V (q)

generates a nice semiflow q(t) = S(t, q(0)) on a dense set of nice functions.Then, it is justified to calculate the (co-variant) Hessian H = HessK(E) as aquadratic form as follows

H(q, ξ) := 〈ξ,K(q)DV (q)∗ξ〉 − 12Dq〈ξ,K(q)ξ〉[V (q)]

Theorem [Liero-M.’13]: Under suitable conditions E is GλC for dK if andonly if H(q, ξ) ≥ λ〈ξ,K(q)ξ〉 on the dense sets for q and ξ.

Assumptions difficult, but necessary conditions follows easily.

For the Hellinger-Kantorivich operator Kα,β(c) and the entropyE(c dx) =

∫ΩE(c(x)) dx

the Hessian can be calculated (as in the Wasserstein case) by using manyintegration by parts. We find the functional inequality

H(c, ξ) =

[α2(A(c)(∆ξ)2 +H(c)|D2ξ|2

)+ αβ

(B1(c)|∇ξ|2 +B2(c)ξ∆ξ

)+ β2B3(c)ξ2

!!≥ λ

[αc|∇ξ|2 + βcξ2

]dx = λ〈ξ,Kα,β(c)ξ〉,

where, using εj(c) = cjE(j)(c), the coefficient are given by

A(c) = ε2(c)+ε1(c)−ε0(c), H(c) = ε1(c)−ε0(c), B1(c) = 32ε1(c)−ε0(c),

B2(c) = −2ε2(c)+ε1(c)−ε0(c), B3(c) = ε2(c)+ 12ε1(c).

• Hellinger-Kantorovich operator K1,4 : ξ 7→ − div(c∇ξ) + 4cξ

• Free energies of the type E(c) =∫

ΩE(c(x))+c(x)V (x) dx

Theorem, in prep.:Geodesic λ-convexity in (M(Ω),HK) for Ω ⊂ Rd closed and convex.

The linear functional c 7→∫

ΩV c dx is geodesically λ-convex, if and only

if the function V(r, x) = r2V (x) is geodesically λ-convex on (CΩ, dCΩ).

Functionals E(c dx) =∫

ΩE(c(x)) dx are geodesically λ-convex

if and only if the function Nλ(ρ, γ) := ρd

γdE(γ2+d

)− λ

2 γ2 satisfies:

(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex

(ii) ρ 7→ Nλ(ρ, γ) is non-increasing for all γ.

γdE(γ2+d

)− λ

2 γ2 satisfies:

(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex

• For p ≥ 1 the functional c 7→∫

Ωcp dx is geodesically 0-convex.

• Logarithmic entropy c 7→∫

ΩλB(c) dx is NOT geodesically λ-convex.

γdE(γ2+d

)− λ

2 γ2 satisfies:

(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex

• For q ∈ ]0, 1[ the functional c 7→∫

(−cq

)dx is geodesically 0-convex,

if and only ifd = 1 and q ∈ [1/3, 1/2] or d = 2 and q = 1/2.

γdE(γ2+d

)− λ

2 γ2 satisfies:

(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex

• total mass M(µ) :=∫

Ωdµ = µ(Ω) is geodesically 2-convex and 2-concave:

M(γ(s)) = (1−s)M(γ(0)) + sM(γ(1))− s(1−s)HK(γ(0), γ(1))2.

Ideas of the proof (w.l.o.g. λ = 0):Necessity of conditions for N follows from Hessian H.

For sufficiency generalize the Brenier/Mc Cann approach:Consider a HK geodesic µ(s) connecting µ0 = c0 dx and µ1 = c1 dx, then

µ(t) = α(t, ·)T (t, ·)#µ0 with

α(t, x) = (1−2tψ(x))2 + t2|∇ψ(x)|2 and

T (t, x) = x− arctan(

t1−2tψ(x)∇ψ(x)

With this we find the geodesic curve in the formµ(t) = c(t, ·) dx with c(t,T (t, x)) = α(t,x)

δ(t,x) c0(x) and δ(t, x) = det(∇T (t, x))

Now the functional readsE(µ(t)) =

∫ΩE(c(t, y)) dy =

∫ΩE(c(t,T (t, x)))δ(t, x) dx =

∫Ωe(t, x) dx

with e(t, x) := E(α(t,x)c0(x)

δ(t,x)

)δ(t, x) dx

Claim: t 7→ e(t, x) is convex for a.a. x ∈ Ω.

Ideas of the proof (w.l.o.g. λ = 0):Necessity of conditions for N follows from Hessian H.

For sufficiency generalize the Brenier/Mc Cann approach:Consider a HK geodesic µ(s) connecting µ0 = c0 dx and µ1 = c1 dx, then

µ(t) = α(t, ·)T (t, ·)#µ0 with

α(t, x) = (1−2tψ(x))2 + t2|∇ψ(x)|2 and

T (t, x) = x− arctan(

t1−2tψ(x)∇ψ(x)

)With this we find the geodesic curve in the formµ(t) = c(t, ·) dx with c(t,T (t, x)) = α(t,x)

δ(t,x) c0(x) and δ(t, x) = det(∇T (t, x))

Now the functional readsE(µ(t)) =

∫ΩE(c(t, y)) dy =

∫ΩE(c(t,T (t, x)))δ(t, x) dx =

∫Ωe(t, x) dx

with e(t, x) := E(α(t,x)c0(x)

δ(t,x)

)δ(t, x) dx

Claim: t 7→ e(t, x) is convex for a.a. x ∈ Ω.

α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(

t1−2tψ

∇ψ), δ = det

)Use optimality of ψ to show δ(t, x) > 0:ψ(x) ≤ 1/2 and D2ψ(x) ≤ a1(ψ, |∇ψ|)I + a2(ψ, |∇ψ|)∇ψ⊗∇ψ

Define new variables γ := (c0α)1/2 and ρ := γδ1/d, then

• e(t, x) = δE(c0α/δ) = N(ρ(t, x), γ(t, x))

• e(t, x) =(ργ

)·D2N(ρ, γ)

)+ DN(ρ, γ)

) N cvx≥ DN(ρ, γ)

) ???≥ 0

α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(

t1−2tψ

∇ψ), δ = det

• e(t, x) =(ργ

)·D2N(ρ, γ)

)+ DN(ρ, γ)

) ???≥ 0

We knowγ =

√positive quadr. polyn. =⇒ γ is convex

∂ρN(ρ, γ) ≤ 0 (and for Wasserstein t 7→(

det(I−tA))1/d is concave)

α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(

t1−2tψ

∇ψ), δ = det

• e(t, x) =(ργ

)·D2N(ρ, γ)

)+ DN(ρ, γ)

) ???≥ 0

Convexity of N and scaling N(s1+2/dρ, sγ) = s2N(ρ, γ) implyadditional monotonicity γ∂γN + (1−4/d2)ρ∂ρN ≥ 0

Convexity γ ≥ 0, concavity ρρ − (1−4/d) γγ ≤ 0 (difficult, more than needed)

Hence, we concludeDN(ρ, γ)

)= ρ∂ρN

(ρρ − (1− 4

d2 ) γγ)

+(γ∂γN + (1− 4

d2 )ρ∂ρN) γγ ≥ 0!!

Overview

1. Motivation

We now consider the gradient system (M(Ω),E,HKα,β) withthe energy functional E(c) =

(−2c(x)1/2 + V (x)c(x)

Assume (r, x) 7→ r2V (x) geodesically λ-convex on CΩ (e.g.V ≡ v0 λ = 8v0/β)

then, for d ∈ 1, 2 the functional E is geodesically λ-convex.

Using DE(c) = −c−1/2 + V , the induced reaction-diffusion equation reads

c = −Kα,β(c)DE(c) = α(

∆c1/2+ div(c∇V ))

+ β(c1/2−cV

)Nonuniqueness and two steady states c ≡ 0 and c∗(x) =

(maxV (x), 0

The Ambrosio-Gigli-Savaré theory [2005] of geodesically λ-convex gradient systems implies

• Existence and uniqueness of solutions for all c(0) ∈M(Ω) withHKα,β(c1(t), c2(t)) ≤ e−λtHKα,β(c1(0), c2(0)) for all t ≥ 0.

• If λ > 0 (e.g. for V ≡ 1), all solutions (even starting at c ≡ 0) satisfyHKα,β(c(t), c∗) ≤ e−λtHKα,β(c(0), c∗)→ 0 for t→∞.

We now consider the gradient system (M(Ω),E,HKα,β) withthe energy functional E(c) =

(−2c(x)1/2 + V (x)c(x)

Assume (r, x) 7→ r2V (x) geodesically λ-convex on CΩ (e.g.V ≡ v0 λ = 8v0/β)

then, for d ∈ 1, 2 the functional E is geodesically λ-convex.

Using DE(c) = −c−1/2 + V , the induced reaction-diffusion equation reads

c = −Kα,β(c)DE(c) = α(

∆c1/2+ div(c∇V ))

+ β(c1/2−cV

)Nonuniqueness and two steady states c ≡ 0 and c∗(x) =

(maxV (x), 0

)2.The Ambrosio-Gigli-Savaré theory [2005] of geodesically λ-convex gradient systems implies

• Existence and uniqueness of solutions for all c(0) ∈M(Ω) withHKα,β(c1(t), c2(t)) ≤ e−λtHKα,β(c1(0), c2(0)) for all t ≥ 0.

• If λ > 0 (e.g. for V ≡ 1), all solutions (even starting at c ≡ 0) satisfyHKα,β(c(t), c∗) ≤ e−λtHKα,β(c(0), c∗)→ 0 for t→∞.

Gradient system (M(Ω),E,HKα,β) withE(c) =

(−2c(x)1/2 + V (x)c(x)

PDE c = −Kα,β(c)DE(c) = α(

∆c1/2+ div(c∇V ))

+ β(c1/2−cV

)Why isn’t the “PDE steady state” c ≡ 0 anequilibrium of the metric gradient system?

Metric slope is |∂E|α,β(0) := lim supµ→0

(E(0)−E(µ)

HKα,β(0, µ).

Using E(0) = 0, E(c dx) ≈ −2∫

Ωc1/2 dx, and HKα,β(0, µ) =

∫Ωc dx

)1/2we find Hellinger-Kantorovich slope |∂E|α,β(0) =

(β|Ω|

)1/2> 0.

Indeed, the metric velocity of solution starting at c ≡ 0 is positive:

HKα,β(0, (1−e−βt/2)2

)= (1−e−βt/2)(4|Ω|/β)1/2 =

(β|Ω|

)1/2t+O(t2).

Gradient system (M(Ω),E,HKα,β) withE(c) =

(−2c(x)1/2 + V (x)c(x)

PDE c = −Kα,β(c)DE(c) = α(

∆c1/2+ div(c∇V ))

+ β(c1/2−cV

)Why isn’t the “PDE steady state” c ≡ 0 anequilibrium of the metric gradient system?

Metric slope is |∂E|α,β(0) := lim supµ→0

(E(0)−E(µ)

HKα,β(0, µ).

Using E(0) = 0, E(c dx) ≈ −2∫

Ωc1/2 dx, and HKα,β(0, µ) =

∫Ωc dx

)1/2we find Hellinger-Kantorovich slope |∂E|α,β(0) =

(β|Ω|

)1/2> 0.

Indeed, the metric velocity of solution starting at c ≡ 0 is positive:

HKα,β(0, (1−e−βt/2)2

)= (1−e−βt/2)(4|Ω|/β)1/2 =

(β|Ω|

)1/2t+O(t2).

Conclusion

The Onsager form u = −K(u)DE(u) allows for easy modeling

The Hellinger-Kantorovich metric allows for an explicit characterization of thedistance and the geodesic curves.

Geodesic convexity with resp. to Hellinger-Kantorovich distance is restrictive• Boltzmann entropy is not GλC•∫

Ωcp dx for p > 1 is GλC with λ = 0 and

∫Ωc dx is G2C.

Construction of solutions for gradient flows for HK is largely open(cf. D. Matthes or V. Laschos)

Thank you for your attention ...

... und Herzlichen Glückwunsch zum Geburtstag für Yann

Liero-M: Gradient structures and geodesic convexity for RDS. PhilTRSA 2013Liero-M-Savaré: Optimal transport in competition with reaction–the HK distance. SIMA 48(4) 2016Liero-M-Savaré: Optimal entropy-transport problems and the HK distance. WIAS#2207, 120 pp.

Conclusion

The Onsager form u = −K(u)DE(u) allows for easy modeling

The Hellinger-Kantorovich metric allows for an explicit characterization of thedistance and the geodesic curves.

Geodesic convexity with resp. to Hellinger-Kantorovich distance is restrictive• Boltzmann entropy is not GλC•∫

Ωcp dx for p > 1 is GλC with λ = 0 and

∫Ωc dx is G2C.

Construction of solutions for gradient flows for HK is largely open(cf. D. Matthes or V. Laschos)

Thank you for your attention ...

... und Herzlichen Glückwunsch zum Geburtstag für Yann

Liero-M: Gradient structures and geodesic convexity for RDS. PhilTRSA 2013Liero-M-Savaré: Optimal transport in competition with reaction–the HK distance. SIMA 48(4) 2016Liero-M-Savaré: Optimal entropy-transport problems and the HK distance. WIAS#2207, 120 pp.

Optimal Transport versus Growth and Decay · Chizat-Peyré-Schmitzer-Vialard3...

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