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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 =
∫Ω
(√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
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 =
∫Ω
(√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” 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 =
∫Ω
(√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)
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 3 (29)
1. Motivation
Cloud motion shows transport, growth, and decay
Rain in France yesterday evening
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 4 (29)
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 5 (29)
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 5 (29)
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 6 (29)
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 6 (29)
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
2. Geodesic curves and dissipation distances
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.
EEE
GAMMA
E(γ(s))
γ(s)
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 7 (29)
2. Geodesic curves and dissipation distances
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))− λ2 s
2dK(γ(0), γ(1))2 is convexfor all such geodesic curves.
EEE
GAMMA
E(γ(s))
γ(s)
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 7 (29)
2. Geodesic curves and dissipation distances
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)
2
=⇒ dK(q(t), qτ (t)) ≤ C(q0)√τ e−λτ t convergence
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 8 (29)
2. Geodesic curves and dissipation distances
How can we characterize dissipation distances defined via K(q)?
dK(q0, q1)2 := inf ∫ 1
0〈G(γ)γ, γ〉 ds
∣∣ 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)
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 9 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ωc2
4c dx dt∣∣ c0 c(t,·)
c1
=∫
Ω
(√c1 −
√c0)2
dx
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
ξ
The Hellinger distance is distinguished: K linear in c !
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 10 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ωc2
4c dx dt∣∣ c0 c(t,·)
c1
=∫
Ω
(√c1 −
√c0)2
dx
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
ξ
The Hellinger distance is distinguished: K linear in c !
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 10 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ωc2
4c dx dt∣∣ c0 c(t,·)
c1
=∫
Ω
(√c1 −
√c0)2
dx
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
ξ
The Hellinger distance is distinguished: K linear in c !
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 10 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ω
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,...
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 11 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ω
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,...
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 11 (29)
2. Geodesic curves and dissipation distances
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
0
∫Ω
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,...
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 11 (29)
2. Geodesic curves and dissipation distances
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
∫ 1
0
∫Ω
(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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 12 (29)
2. Geodesic curves and dissipation distances
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
∫ 1
0
∫Ω
(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
= 4β
∫Ω
(√µ0 −
õ1
)2dx
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 12 (29)
2. Geodesic curves and dissipation distances
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
∫ 1
0
∫Ω
(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
= 4β
∫Ω
(√µ0 −
õ1
)2dx
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 12 (29)
2. Geodesic curves and dissipation distancesinitial measureµ1 = χ[1,2]
final measureµ2 = δ0
left: α = 0Hellinger geodesics((1−s)√µ0+s
õ1
)2
middle: β = 0Wassersteintransport
right: αβ > 0Hellinger-Kantorovichgeodesic(transport andabsorp./gener.)
Hellinger-Kakutani Wasserstein-Kantorovich Hellinger-Kantorovich
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 13 (29)
2. Geodesic curves and dissipation distancesinitial measureµ1 = χ[1,2]
final measureµ2 = δ0
left: α = 0Hellinger geodesics((1−s)√µ0+s
õ1
)2middle: β = 0Wassersteintransport
right: αβ > 0Hellinger-Kantorovichgeodesic(transport andabsorp./gener.)
Hellinger-Kakutani Wasserstein-Kantorovich Hellinger-Kantorovich
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 13 (29)
2. Geodesic curves and dissipation distancesinitial measureµ1 = χ[1,2]
final measureµ2 = δ0
left: α = 0Hellinger geodesics((1−s)√µ0+s
õ1
)2middle: β = 0Wassersteintransport
right: αβ > 0Hellinger-Kantorovichgeodesic(transport andabsorp./gener.)
Hellinger-Kakutani Wasserstein-Kantorovich Hellinger-Kantorovich
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 13 (29)
2. Geodesic curves and dissipation distances
For α = 1 and β = 4 move µ0 =30∑k=0
δ(1,k/15) to µ1 = 31 δ(0,0)
0.0
0.5
1.0
0.0
0.5
1.0
1.5
2.0
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0
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2
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 14 (29)
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
3. Characterizations of HKα,β
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
2 = 0
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))?
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 15 (29)
3. Characterizations of HKα,β
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
2 = 0
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))?
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 15 (29)
3. Characterizations of HKα,β
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))
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 16 (29)
3. Characterizations of HKα,β
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
0
( 1
β
a(s)2
a(s)+a(s)
α|x(s)|2
)ds
... 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
)2 =
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. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 17 (29)
3. Characterizations of HKα,β
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
0
( 1
β
a(s)2
a(s)+a(s)
α|x(s)|2
)ds
... 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
)2 =
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. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 17 (29)
3. Characterizations of HKα,β
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
0
( 1
β
a(s)2
a(s)+a(s)
α|x(s)|2
)ds
... 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
)2 =
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. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 17 (29)
3. Characterizations of HKα,β
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 18 (29)
3. Characterizations of HKα,β
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 19 (29)
3. Characterizations of HKα,β
Characterization of HK via entropy regularization
HK(µ0, µ1)2 = infη
(∫Ω
λB( dΠ0
#η
dµ0
)dµ0 +
∫Ω
λB( dΠ1
#η
dµ1
)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
( r20
ϑ
)+ λB
( r21
ϑ
)+ ϑc(r)
)with
Boltzmann’s entropy function λB(z) = z log z−z+1 (i.e.λ′B(z) = log z)
and the cost function c(r) =
2 log
(1
cos r
)for r < π/2,
∞ otherwise.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 20 (29)
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
4. Geodesic λ-convexity for HK
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 21 (29)
4. Geodesic λ-convexity for HK
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
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 21 (29)
4. Geodesic λ-convexity for HK
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 22 (29)
4. Geodesic λ-convexity for HK
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 22 (29)
4. Geodesic λ-convexity for HK
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) =
∫Ω
[α2(A(c)(∆ξ)2 +H(c)|D2ξ|2
)+ αβ
(B1(c)|∇ξ|2 +B2(c)ξ∆ξ
)+ β2B3(c)ξ2
]dx
!!≥ λ
∫Ω
[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).
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 23 (29)
4. Geodesic λ-convexity for HK
• 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
ρd
)− λ
2 γ2 satisfies:
(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex
(ii) ρ 7→ Nλ(ρ, γ) is non-increasing for all γ.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 24 (29)
4. Geodesic λ-convexity for HK
• 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
ρd
)− λ
2 γ2 satisfies:
(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex
(ii) ρ 7→ Nλ(ρ, γ) is non-increasing for all γ.
• 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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 24 (29)
4. Geodesic λ-convexity for HK
• 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
ρd
)− λ
2 γ2 satisfies:
(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex
(ii) ρ 7→ Nλ(ρ, γ) is non-increasing for all γ.
• For q ∈ ]0, 1[ the functional c 7→∫
Ω
(−cq
)dx is geodesically 0-convex,
if and only ifd = 1 and q ∈ [1/3, 1/2] or d = 2 and q = 1/2.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 24 (29)
4. Geodesic λ-convexity for HK
• 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
ρd
)− λ
2 γ2 satisfies:
(i) Nλ : ]0,∞[2 → R ∪ ∞ is convex
(ii) ρ 7→ Nλ(ρ, γ) is non-increasing for all γ.
• 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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 24 (29)
4. Geodesic λ-convexity for HK
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 25 (29)
4. Geodesic λ-convexity for HK
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 25 (29)
4. Geodesic λ-convexity for HK
α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(
t1−2tψ
∇ψ), δ = det
(∇T
)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)
(ργ
)+ DN(ρ, γ)
(ργ
) N cvx≥ DN(ρ, γ)
(ργ
) ???≥ 0
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 26 (29)
4. Geodesic λ-convexity for HK
α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(
t1−2tψ
∇ψ), δ = det
(∇T
)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)
(ργ
)+ DN(ρ, γ)
(ργ
) N cvx≥ DN(ρ, γ)
(ργ
) ???≥ 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)
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 26 (29)
4. Geodesic λ-convexity for HK
α = (1−2tψ)2 + t2|∇ψ|2, T = x− arctan(
t1−2tψ
∇ψ), δ = det
(∇T
)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)
(ργ
)+ DN(ρ, γ)
(ργ
) N cvx≥ DN(ρ, γ)
(ργ
) ???≥ 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)
(ργ
)= ρ∂ρN
(ρρ − (1− 4
d2 ) γγ)
+(γ∂γN + (1− 4
d2 )ρ∂ρN) γγ ≥ 0!!
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 26 (29)
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
5. A fast-diffusion equation with growth and decay
We now consider the gradient system (M(Ω),E,HKα,β) withthe energy functional E(c) =
∫Ω
(−2c(x)1/2 + V (x)c(x)
)dx.
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→∞.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 27 (29)
5. A fast-diffusion equation with growth and decay
We now consider the gradient system (M(Ω),E,HKα,β) withthe energy functional E(c) =
∫Ω
(−2c(x)1/2 + V (x)c(x)
)dx.
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→∞.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 27 (29)
5. A fast-diffusion equation with growth and decay
Gradient system (M(Ω),E,HKα,β) withE(c) =
∫Ω
(−2c(x)1/2 + V (x)c(x)
)dx.
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) =
(4β
∫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).
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 28 (29)
5. A fast-diffusion equation with growth and decay
Gradient system (M(Ω),E,HKα,β) withE(c) =
∫Ω
(−2c(x)1/2 + V (x)c(x)
)dx.
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) =
(4β
∫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).
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 28 (29)
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 29 (29)
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.
A. Mielke, Optimal transport vs growth/decay, Paris, 11. Jan. 2017 29 (29)
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