on dissipation distances for reaction-diffusion equations ... · hellinger-kantorovich distance...

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

http://www.wias-berlin.de

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


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


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)


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)⇠


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


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


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


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


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


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


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

↵

�

⇡



13


only absorp./gen.



✓(t)

t

a0a1

> 1

a0a1

< 1a0

a1

|x0�x1| <r

↵

�

⇡

|x0�x1| >r

↵

�

⇡



14


only absorp./gen.



✓(t)

t

a0a1

> 1

a0a1

< 1a0

a1

|x0�x1| <r

↵

�

⇡

|x0�x1| >r

↵

�

⇡


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| ⇡


The cone distance

16

x0 x1

Identify all points in ⌦⇥ {0}

R↵� :=q

�4↵ |x0�x1| ⇡


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


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

)


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


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


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


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)

on dissipation distances for reaction-diffusion equations ... · hellinger-kantorovich distance...

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