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Weierstrass Institute for !Applied Analysis and Stochastics
Mohrenstr. 39 · 10117 Berlin · Tel. +49 30 203 72 0 · www.wias-berlin.de · July 1st, 2014
Matthias Liero, joint work with Alexander Mielke, Giuseppe Savaré
1
On dissipation distances for reaction-diffusion equations — the Hellinger-Kantorovich distance
Hellinger-Kantorovich distance
OutlineIntroduction to Hellinger-Kantorovich distance !
!Wasserstein distance on the space of probability measures
!Dissipation distance induced by reaction-diffusion problem
!Example: Optimal relocation of mass points
!Main results
!
Thorough discussion in Giuseppe’s talk
2
Hellinger-Kantorovich distance
The Wasserstein distance!Kantorovich-Wasserstein distance on space of prob. meas.
3
W2(µ0, µ1)2 = min
n
ZZ
Rd⇥Rd
|x�y|2 d⌘(x, y)o
⌘(·⇥ ⌦) = µ0, ⌘(⌦⇥ ·) = µ1Transport plan
!Equivalent dynamical formulation (Benamou-Brenier-2000)
W2(µ0, µ1)2 = min
n
Z 1
0
Z
Rd
|vt(x)|2 dµt(x)dto
@tµt +r · (vtµt) = 0continuity equation
Hellinger-Kantorovich distance
The Wasserstein distance
4
gµ(µ, µ) =
Z
Rd
r⇠µ ·r⇠µ dµ µ+r · (µr⇠µ) = 0
Otto’s Riemannian interpretation
Wasserstein gradient flows
@tu = �gradWE(u) @tu = r · (uE00(u)ru)
E(u) =Z
⌦E(u)dx gradWE(u) = �r · (ur �E
�u )
Wasserstein gradient
see Ambrosio-Gigli-Savaré-2005
µ = �r · (r⇠µµ)
Hellinger-Kantorovich distance
adjust effect of diffusion/reaction
Generalizations of Wasserstein
5
�r · (µr⇠)
Mielke-2011: Onsager operators
K↵�(u)⇠ = �r · (↵ur⇠) + �u⇠
General case too hard, consider
Understanding geometry gives results for scalar RD equation
@tu = �gradK↵�E(u)
@tu = r · (↵uE00(u)ru)� �uE0(u)
Can we characterize the associated reaction-diffusion distance?(e.g. reversible mass-action kinetics, semiconductor eqns.)
K(u)⇠ = �r · (M(u)r⇠) +H(u)⇠
Hellinger-Kantorovich distance
Competition between reaction/diffusion
6
Reaction-diffusion distance
vt = ↵r⇠t wt = �⇠tBenamou-Brenier-trick
inf-convolution of Wasserstein-Kantorovich and Hellinger distance
Competition between optimal transport and absorption/generation
D0�(µ0, µ1)2 =
4
�
Z
⌦
�pf0�
pf1�2d�D↵0(µ0, µ1) =
1p↵W2(µ0, µ1)
D↵�(µ0, µ1)2 = inf
n
Z 1
0
Z
⌦
⇣
↵|r⇠t|2 + �⇠2t
⌘
dµtdto
generalized cont. eq. @tµt +r · (↵r⇠tµt) = �⇠tµt
Hellinger-Kantorovich distance
Competition between reaction/diffusion
7
Reaction-diffusion distance
vt = ↵r⇠t wt = �⇠tBenamou-Brenier-trick
inf-convolution of Wasserstein-Kantorovich and Hellinger distance
Competition between optimal transport and absorption/generation
D0�(µ0, µ1)2 =
4
�
Z
⌦
�pf0�
pf1�2d�D↵0(µ0, µ1) =
1p↵W2(µ0, µ1)
D↵�(µ0, µ1)2 = inf
n
Z 1
0
Z
⌦
⇣ |vt|2
↵+
w2t
�
⌘
dµtdto
generalized cont. eq. @tµt +r · (vtµt) = wtµt
Hellinger-Kantorovich distance
Two mass points
8
µ0 = a0�x0 µ1 = a1�x1Optimal relocation
x0
x1
c(s)
x(s)
a0
a1
b0(s)
b1(s)Naive approach
Hellinger-Kantorovich distance
Two mass points!Consider only curves of the form
9
µt
= b0(t)�x0 + c(t)�x(t) + b1(t)�x1
x0
x1
c(s)
x(s)
a0
a1
b0(s)
b1(s)
x(0) = x0, x(1) = x1
b0(0) + c(0) = a0
b1(1) + c(1) = a1
b1(0) = 0
b0(1) = 0
Initial and final conditions
wt(xi) =bi(t)
bi(t), wt(x(t)) =
c(s)
c(s), vt(x(t)) = x(t)
Solve gen. continuity equation @tµt +r · (vtµt)� wtµt = 0
Hellinger-Kantorovich distance
Two mass points!Plug into action functional and solve Euler-Lagrange equations
10
b0(t) = (1�t)2b0(0), b1(t) = t2b1(1)
Absorption/Generation
c(t) = (1�t)2c(0) + t2c(1) + 2t(1�t)pc(0)c(1) cos
�R↵�
�
x(t) =�1�✓(t)
�x0 + ✓(t)x1
TransportR↵� :=
q�4↵ |x0�x1| ⇡
✓(s) =1
R↵�arctan
⇣ sp
c(1) sin(R↵�)
(1�s)pc(0) + s
pc(1) cos(R↵�)
⌘Speed of transport determined by
Hellinger-Kantorovich distance
Two mass points
11
A(b0, b1, c, x) =4
�
n
b0(0) + c(0)
| {z }
a0
+ b1(1) + c(1)
| {z }
a1
�2
p
c(0)c(1) cos(R↵�)
o
Minimize with respect to initial and final distribution of mass
R↵� >⇡
2
minA =4
�(a0 + a1)
b0(0) = a0, b1(1) = a1, c(0) = c(1) = 0
Make and as small as possiblec(0) c(1)
Make and as big as possiblec(0) c(1)
c(0) = a0, c(1) = a1, b0(0) = b1(1) = 0
minA =
4
�
�a0 + a1 � 2
pa0a1 cos(R↵�)
�
R↵� <⇡
2
Hellinger-Kantorovich distance
Observations!Sharp threshold
12
only (mixed) transport
only absorp./gen.
!Speed of transport depends on initial and final mass
!Mass is not conserved along transport
✓(t)
t
a0a1
> 1
a0a1
< 1a0
a1
|x0�x1| <r
↵
�
⇡
|x0�x1| >r
↵
�
⇡
Hellinger-Kantorovich distance
Observations!Sharp threshold
13
only (mixed) transport
only absorp./gen.
!Speed of transport depends on initial and final mass
!Mass is not conserved along transport
✓(t)
t
a0a1
> 1
a0a1
< 1a0
a1
|x0�x1| <r
↵
�
⇡
|x0�x1| >r
↵
�
⇡
Hellinger-Kantorovich distance
Observations!Sharp threshold
14
only (mixed) transport
only absorp./gen.
!Speed of transport depends on initial and final mass
!Mass is not conserved along transport
✓(t)
t
a0a1
> 1
a0a1
< 1a0
a1
|x0�x1| <r
↵
�
⇡
|x0�x1| >r
↵
�
⇡
Hellinger-Kantorovich distance
The cone distance
15
⌦
x0x1
Consider fiber in each point [0,1[
gives distance on cone space over ⌦
D(x0, a0;x1, a1) :=4
�
�a0 + a1 � 2
pa0a1 cos(R↵�)
�
R↵� :=q
�4↵ |x0�x1| ⇡
Hellinger-Kantorovich distance
The cone distance
16
x0 x1
Identify all points in ⌦⇥ {0}
R↵� :=q
�4↵ |x0�x1| ⇡
Hellinger-Kantorovich distance
The cone distance
17
C⌦ =�⌦⇥[0,1[
�.�⌦⇥{0}
�
0C
R↵� :=q
�4↵ |x0�x1| ⇡
pa0
pa1
[x(s), a(s)]
Law of cosines:
d
Cone
([x
0
, a
0
], [x
1
, a
1
])
2
=
(a
0
+ a
1
� 2
pa
0
a
1
cos(R↵�) R↵� < ⇡
(
pa
0
+
pa
1
)
2
R↵� � ⇡
1
Solutions of Euler-Lagrange eqns!are geodesics in cone space
Hellinger-Kantorovich distance
Two mass points
18
cost of transport given by4
�
d
Cone
([x
0
, a
0
], [x
1
, a
1
])
2
=
4
�
�a
0
+ a
1
� 2
pa
0
a
1
cos(
p�/(4↵)|x
0
�x
1
|)�
|x0�x1|↵�
pa0
pa1
0C
|x0�x1| <r
↵
�
⇡Transport:
|x0�x1|↵�
pa0
pa1
0C
|x0�x1| >r
↵
�
⇡Absorption/generation:
cost of absorption/generation given by
4
�
d
Cone
([x0
, a
0
],0C)2 +
4
�
d
Cone
(0C, [x1
, a
1
])2 =4
�
�a
0
+ a
1
)
Hellinger-Kantorovich distance
Extension to general measures
19
Connection to measures on ⌦ provided by projection
P : M�0(C⌦) ! M�0(⌦)Z
⌦'(x) dP⌫i =
Z
C⌦
ba(z)'(bx(z)) d⌫i(z)
⌫0 = �0⌦ + �[x0,a0] ⌫1 = �0⌦ + �[x1,a1]
|x0�x1| <r
↵
�
⇡
|x0�x1| >r
↵
�
⇡
Optimal relocation in induced by optimal transport in⌦ C⌦
Define HK(µ0, µ1) =2p�min
�WC⌦(⌫0, ⌫1) |µi = P⌫i
Hellinger-Kantorovich distance
Main result
20
Theorem [L.-Mielke-Savaré-2014]
M�0(⌦)! defines distance on!! complete, separable metric space!! metrizes the weak topology!! minimizer exists!! geodesic curves induced by geodesics in!! equivalent to dynamical formulation
C⌦
HK(µ0, µ1)2 = min
⌫2H(µ0,µ1)
ZZ
C⌦⇥C⌦
dCone(z0, z1)2 d⌫
⌫ = H(µ0, µ1) () P(⌫i) = µi
Hellinger-Kantorovich distance
Another equivalent characterization
21
Theorem [L.-Mielke-Savaré-2014]
F (z) = z log z � z + 1
HK(µ0, µ1)2 =
4
�
minn
Z
⌦F
� d⌘0dµ0
�
dµ0 +
Z
⌦F
� d⌘1dµ1
�
dµ1 +
ZZ
⌦⇥⌦�↵�(|x0�x1|)d⌘
o
�↵�(R) =
(�2 log
⇣cos
�p�/(4↵)R
�⌘R < ⇡
p↵/�
1 R � ⇡p
↵/�
Cost function
Boltzmann function
Equivalent formulation
⌘i = P i#⌘ ⌧ µiMarginals
Hellinger-Kantorovich distance
Geodesic curves
22
3. Geodesic curves and dissipation distances
Move µ0 =30!k=0
δ(1,k/15) to µ1 = 31 δ(0,0)
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
0.0
0.5
1.0
0.00.5
1.01.5
2.0
0
1
2
A. Mielke, Gradient Structures and Diss.Dist. for RDS, Wien, 2. Juni 2014 26 (30)