Optimal Transport for a Computer Programmer's Point of View

Mathématiques - Informatique

Optimal Transport

From a Computer Perspective

Summer School « Geometric Measure Theory and

Calculus of Variations », Grenoble, June 2015

Bruno Lévy

ALICE Géométrie & Lumière

CENTRE INRIA Nancy Grand-Est

Part. 1. Goals and Motivations - Understanding

Part. 2. Optimal Transport Elementary intro.

Part. 3. The Semi-Discrete Case

Part. 4. Understanding Going

Part. 5. Concluding Words

OVERVIEW

OVERVIEW

Cédric Villani

Optimal Transport Old & New

Topics on Optimal Transport

Yann Brenier

The polar factorization theorem

(Brenier Transport)

OVERVIEW

June 2015

Institut Fourier

March 2015

Bonn

Febr 2015

BIRS (Canada)

Febr 2015

LJLL

Goals and Motivations

1

Part. 1 Optimal TransportMeasuring distances between functions



Part. 1 Optimal TransportMeasuring distances between function

Part. 1 Optimal TransportInterpolating functions












Part. 1 Optimal Transport

?

T


?

T

distance between the shapes


while preserving mass and minimizing the effort ?

?

T



?

T



?

T


OT=

Yann Brenier:

it can be replaced with the Monge-Ampère


OT=

Yann Brenier:


New ways of simulating physics with a computer


OT=

Yann Brenier:

Laplace operator is used in a PDE,



Fast Fourier Transform


OT=

Yann Brenier:

Laplace operator is used in a PDE,



Fast Fourier Transform Fast OT algo. ???


Gaspard Monge - 1784


Gaspard Monge geometry and light

ANR TOMMI Workshop


[Castro, Merigot, Thibert 2014]

Optimal transport

geometry and light

[Caffarelli, Kochengin, and Oliker 1999]

Part. 1 Optimal Transport Image Processing

Barycenters / mixing textures Video-style transfer

[Nicolas Bonneel, Julien Rabin, Gabriel

Peyre, Hanspeter Pfister][Nicolas Bonneel, Kalyan Sunkavalli, Sylvain

Paris, Hanspeter Pfister]


Optimal transport - geometry and light

[Chwartzburg, Testuz, Tagliasacchi, Pauly, SIGGRAPH 2014]

Part. 1. Motivations

Discretization of functionals involving the Monge-Ampère operator,

Benamou, Carlier, Mérigot, Oudet

arXiv:1408.4536

The variational formulation of the Fokker-Planck equation

Jordan, Kinderlehrer and Otto

SIAM J. on Mathematical Analysis

Geostrophic current


Optimal transport E.U.R.

The data


The millenium simulation project, Max Planck Institute fur Astrophysik

pc/h : parsec (= 3.2 light years)



The universal swimming pool



let there be light !

The millenium simulation project,

Max Planck Institute fur Astrophysik








In 2002: 5 hours of computation

on a super-computer for 5000 points

with a O(n3) algorithm.







In 2002: 5 hours of computation

on a super-computer for 5000 points

with a O(n3) algorithm.

Can we do the computation with

1 000 000 points ?

Optimal Transportan elementary introduction

2

Part. 2 Optimal Transport problem

(X; ) (Y; )

Two measures , such that d (x) = d (x)X Y


A map T is a transport map between and if

(T-1(B)) = (B) for any Borel subset B of Y

(X; ) (Y; )



(T-1(B)) = (B) for any Borel subset B

B

(X; ) (Y; )




BT-1(B)

(X; ) (Y; )




(or = T# the pushforward of )

(X; ) (Y; )


problem:

Find a transport map T that minimizes C(T) = X || x T(x) ||2 d (x)

(X; ) (Y; )


problem:


Difficult to study

If has an atom (isolated Dirac),

it can only be mapped to another Dirac

(T needs to be a map)


problem:


Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3


problem:




problem:




problem:



The infimum is never realized by a map, need for a relaxation

Part. 2 Optimal Transport Kantorovich

problem:


(1942):

Find a measure defined on X x Y

such that x in X d (x,y) = d (y)

and y in Y d (x,y) = d (x)

that minimizes X x Y || x y ||2 d (x,y)


problem:






(x,y

How much sand goes from x to y


problem:






Everything that is

transported from x


problem:






Everything that is

transported to y


Transport plan example 1/4 : translation of a segment


Transport plan example 1/4 : translation of a segment


Transport plan example 2/4 : spitting a segment





Transport plan example 3/4 : splitting a Dirac into two Diracs


Transport plan example 3/4 : splitting a Dirac into two Diracs

(No transport map)


Transport plan example 4/4 : splitting a Dirac into two segments


Transport plan example 4/4 : splitting a Dirac into two segments

(No transport map)

Part. 2 Optimal Transport Duality


Duality is easier to understand with a discrete version

Part. 2 Optimal Transport Duality(DMK):

Min <c, >

P1 = u

s.t. P2 = v


Min <c, >

P1 = u

s.t. P2 = v

=

11

12

1n

22

2n

mn

IRmn


Min <c, >

P1 = u

s.t. P2 = v

=

11

12

1n

22

2n

mn

c =

c11

c12

c1n

c22

c2n

cmn

IRmnIRmn


Min <c, >

P1 = u

s.t. P2 = v

=

11

12

1n

22

2n

mn

c =

c11

c12

c1n

c22

c2n

cmncij = || xi yj ||2

IRmnIRmn


Min <c, >

P1 = u

s.t. P2 = v

0

=

11

12

1n

22

2n

mn

c =

c11

c12

c1n

c22

c2n


mn X m

IRmnIRmn


Min <c, >

P1 = u

s.t. P2 = v

=

11

12

1n

22

2n

mn

c =

c11

c12

c1n

c22

c2n


mn X n

mn X m

IRmnIRmn


Min <c, >

P1 = u

s.t. P2 = v

=

11

12

1n

22

2n

mn

c =

c11

c12

c1n

c22

c2n

cmn

IRmnIRmn

mn X n

mn X m


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>

(DMK):

Min <c, >

P1 = u

s.t. P2 = v< u, v > denotes the dot product between u and v


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>

Remark: Sup[ L( , ) ] = < c, > if P1 = u and P2 = v IRm

IRn

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>


IRn

(DMK):

Min <c, >

P1 = u

s.t. P2 = v

0


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>


IRn

Consider now: Inf [ Sup[ L( , ) ] ]IRm

IRn

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>


IRn

Consider now: Inf [ Sup[ L( , ) ] ] = Inf [ < c, > ] IRm

IRn P1 = u

P2 = v

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


Consider L( , ) = <c, > - < , P1 - u> - < , P2 - v>


IRn

Consider now: Inf [ Sup[ L( , ) ] ] = Inf [ < c, > ] (DMK)IRm

IRn0

P1 = u

P2 = v

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


IRm

IRn

Inf [ Sup[ <c, > - < , P1 - u> - < , P2 - v> ] ]

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


IRm

IRn0

Inf [ Sup[ <c, > - < , P1 - u> - < , P2 - v> ] ]

IRm

IRn

Sup[ Inf[ <c, > - < , P1 - u> - < , P2 - v> ] ]

Exchange Inf Sup

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


IRm

IRn

Inf [ Sup[ <c, > - < , P1 - u> - < , P2 - v> ] ]

IRm

IRn

Sup[ Inf[ <c, > - < , P1 - u> - < , P2 - v> ] ]

IRm

IRn

0

Sup[ Inf[ < ,c-P1t P2

t > + < ,u> + < , v> ] ]

Exchange Inf Sup

Expand/Reorder/Collect

(DMK):

Min <c, >

P1 = u

s.t. P2 = v

0


IRm

IRn

Inf [ Sup[ <c, > - < , P1 - u> - < , P2 - v> ] ]

IRm

IRn

Sup[ Inf[ <c, > - < , P1 - u> - < , P2 - v> ] ]

IRm

IRn


t > + < ,u> + < , v> ] ]

Exchange Inf Sup

Expand/Reorder/Collect

Interpret

(DMK):

Min <c, >

P1 = u

s.t. P2 = v


IRm

IRn


t > + < ,u> + < , v> ] ]

Interpret

Sup[ < ,u> + < , v> ] IRm

IRn

P1t + P2

t c

(DMK):

Min <c, >

P1 = u

s.t. P2 = v

(DDMK)


IRm

IRn


t > + < ,u> + < , v> ] ]

Interpret

Sup[ < ,u> + < , v> ] IRm

IRn

P1t + P2

t c i + j cij (i,j)

A

(DMK):

Min <c, >

P1 = u

s.t. P2 = v

(DDMK)

Part. 2 Optimal Transport Kantorovich dual


such that x in X d (x,y) = d (x)



Find two functions in L1( ) and in L1( )

Such that for all x,y, (x) + y ||2

that maximize X d + Y d








that maximize X d + Y d

Your point of view:

Try to minimize transport cost








that maximize X d + Y d Try to maximize transport price

Your point of view:









that maximize X (x)d + Y (y)d

What they charge for loading at x

Your point of view:










What they charge for loading at x What they charge for unloading at y

Your point of view:










What they charge for loading at x What they charge for unloading at y

Price (loading + unloading) cannot

be greater than transport cost

(else you do the job yourself)

Your point of view:


Part. 2 Optimal Transport c-conjugate functions








If we got two functions and that satisfy the constraint

Then it is possible to obtain a better solution by replacing with c defined by:

For all y, c(y) = inf x in X ½|| x y ||2 - (y)





If we got two functions and that satisfy the constraint

Then it is possible to obtain a better solution by replacing with c defined by:

For all y, c(y) = inf x in X ½|| x y ||2 - (y)

c is called the c-conjugate function of

If there is a function such that = c then is said to be c-concave

If is c-concave, then cc =


Find a c-concave function

that maximizes X (x)d + Yc(y)d




is called a

Part. 2 Optimal Transport c-subdifferential

What about our initial problem ?

is called a




is called a





Proof: see OTON, chap. 10.



Heuristic argument (at the beginning of the same chapter):



















w




-w







{

grad (x) with (x) := (½ x2- (x))

Part. 2 Optimal Transport convexity



{

grad (x) with (x) := (½ x2- (x))

is convex




{

grad (x) with (x) := (½ x2- (x))

is convex




{

grad (x) with (x) := (½ x2- (x))

is convex

Part. 2 Optimal Transport no collision

If T(.) exists, then

T(x) = x grad (x) = grad (½ x2- (x) ) {grad (x)



is convex






is convex






is convex

Part. 2 Optimal Transport Monge-Ampere

What about our initial problem ? If T(.) exists, then one can show that:

T(x) = x grad (x) = grad (½ x2- (x) ) {

for all borel set A, A d = T(A) (|JT|) d (change of variable)

Jacobian of T (1st order derivatives)



grad (x) with (x) := (½ x2- (x))


What about our initial problem ? If T(.) exists, then one can show that:

T(x) = x grad (x) = grad (½ x2- (x) ) {

for all borel set A, A d = T(A) (|JT|) d = T(A) (H ) d

Det. of the Hessian of (2nd order derivatives)



grad (x) with (x) := (½ x2- (x))


What about our initial problem ?

T(x) = x grad (x) = grad (½ x2- (x) ) {

grad (x) with (x) := (½ x2- (x))

When and have a density u and v, (H (x)). v(grad (x)) = u(x)Monge-Ampère

equation

for all borel set A, A d = T(A) (|JT|) d = T(A) (H ) d



Part. 2 Optimal Transport summary




(Kantorovich formulation, dual, c-convex functions)




Monge-Ampère equation (When and have a density u and v resp.)

Solve (H (x)). v(grad (x)) = u(x)




Brenier, Mc Cann, Trudinger: The optimal transport map is then given by:

T(x) = grad (x)

Solve (H (x)). v(grad (x)) = u(x) Monge-Ampère equation (When and have a density u and v resp.)

Part. 2 Optimal Transport Isoperimetric inequality

For a given volume,

ball is the shape that minimizes border area


Vol(B2n)1/n ( f n/(n-1))(n-1)/n

L1 Sobolev inegality: Given f: IRn sufficiently regular

Explanation in [Dario Cordero Erauquin] course notes


n Vol(B2n)1/n ( f n/(n-1))(n-1)/n




Vol(B2n)1/n ( f n/(n-1))(n-1)/n




Vol(B2n)1/n ( f n/(n-1))(n-1)/n


Consider a compact set such that Vol( ) = Vol(B23)

and f = the indicatrix function of


Vol(B2n)1/n ( f n/(n-1))(n-1)/n




Vol Vol(B23)1/3 Vol(B2

3)2/3


Vol(B2n)1/n ( f n/(n-1))(n-1)/n




Vol Vol(B23)1/3 Vol(B2

3)2/3

Vol = Vol 23)


Vol(B2n)1/n ( f n/(n-1))(n-1)/n

L1 Sobolev inegality: a proof with OT [Gromov]


Vol(B2n)1/n ( f n/(n-1))(n-1)/n


We suppose w.l.o.g. that f n/(n-1) = 1


Vol(B2n)1/n ( f n/(n-1))(n-1)/n


There exists an optimal transport T = grad between

f n/(n-1)(x)dx and 1B2n/Vol(B2

n)dx



Vol(B2n)1/n ( f n/(n-1))(n-1)/n




n)dx


Monge-Ampère equation: Vol(B2n) fn/(n-1)(x) = det Hess


Vol(B2n)1/n ( f n/(n-1))(n-1)/n




n)dx



Arithmetico-geometric ineq: det (H) 1/n


Vol(B2n)1/n ( f n/(n-1))(n-1)/n




n)dx




det (Hess ) 1/n )/n


Vol(B2n)1/n ( f n/(n-1))(n-1)/n




n)dx




det (Hess ) 1/n )/n

det (Hess ) 1/n / n


Vol(B2n)1/n ( f n/(n-1))(n-1)/n



det (Hess ) 1/n )/nMonge-Ampère equation:

Vol(B2n) fn/(n-1)(x) = det Hess


Vol(B2n)1/n ( f n/(n-1))(n-1)/n



Vol(B2n) = Vol(B2

n) f n/(n-1) = f Vol(B2n) f 1/(n-1)




Vol(B2n)1/n ( f n/(n-1))(n-1)/n



Vol(B2n) = Vol(B2

n) f n/(n-1) = f Vol(B2n) f 1/(n-1)




Vol(B2n)1/n ( f n/(n-1))(n-1)/n



Vol(B2n) = Vol(B2

n) f n/(n-1) = f Vol(B2n) f 1/(n-1)

= - grad f . grad




Vol(B2n)1/n ( f n/(n-1))(n-1)/n



Vol(B2n) = Vol(B2

n) f n/(n-1) = f Vol(B2n) f 1/(n-1) 1/n

= - grad f . grad | grad f | (T = grad B2n )




Vol(B2n)1/n ( f n/(n-1))(n-1)/n



Vol(B2n) = Vol(B2

n) f n/(n-1) = f Vol(B2n) f 1/(n-1)

= - grad f . grad | grad f | (T = grad B2n )

Vol(B2n)1/n



Semi-Discrete Optimal Transport

3

Part. 3 Optimal Transport how to program ?

Continuous

(X; ) (Y; )


Continuous

Semi-discrete

(X; ) (Y; )


Continuous

Semi-discrete

Discrete

(X; ) (Y; )


Continuous

Semi-discrete

Discrete

(X; ) (Y; )

Part. 3 Optimal Transport semi-discrete

Xc (x)d + Y (y)d

Supc

(DMK)

(X; ) (Y; )

Part. 3 Optimal Transport semi-discrete (X; ) (Y; )

j (yj) vj

Xc (x)d + Y (y)d

Supc

(DMK)


Xc (x)d + Y (y)d

Supc

(DMK)

j (yj) vj


Xc (x)d + Y (y)d

Supc

(DMK)

X inf yj Y [ || x yj ||2 - (yj) ] d

j (yj) vj


Xc (x)d + Y (y)d

Supc

(DMK)

j Lag (yj) || x yj ||2 - (yj) d

X inf yj Y [ || x yj ||2 - (yj) ] d

j (yj) vj


G( ) = j Lag (yj) || x yj ||2 - (yj) d + j (yj) vjSup

c

(DMK)

Where: Lag (yj) = { x | || x yj ||2 (yj) < || x yj ||2 - (yj ) }



c

(DMK)


Laguerre diagram of the yj

(with the L2 cost || x y ||2 used here, Power diagram)



c

(DMK)




Weight of yj in the power diagram



c

(DMK)





is determined by the

weight vector [ (y1) (y2 (ym)]



c

(DMK)





is determined by the

weight vector [ (y1) (y2 (ym)]

For all weight vector, is c-concave

Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) }

Part. 3 Power Diagrams

Power diagram: Pow(xi) = { x | d2(x,xi) i < d2(x,xj) j }

Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) }




Theorem: (direct consequence of MK duality

alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ):

Given a measure with density, a set of points (yj), a set of positive coefficients vj

such that vj d (x), it is possible to find the weights W = [ (y1) (y2 (ym)]

such that the map TSW is the unique optimal transport map

between and = vj (yj)








Proof: G( ) = j Lag (yj) || x yj ||2 - (yj) d + j (yj) vj

Is a concave function of the weight vector [ (y1) (y2 (ym)]








Proof: G( ) = j Lag (yj) || x yj ||2 - (yj) d + j (yj) vj

Is a concave function of the weight vector [ (y1) (y2 (ym)]

Part. 3 Optimal Transport the AHA paper

Idea of the proof

Consider the function fT(W) = (|| x T(x) ||2 (T(X))) d (x)

The (unknown) weights W = [ (y1) (y2 (ym)]


Idea of the proof


T : an arbitrary but fixed assignment.


Idea of the proof




Idea of the proof




Idea of the proof




Idea of the proof


W

fT(W)

fT(W)


Idea of the proof


W

fT(W)

fT(W)


Idea of the proof


fT(W) is linear in W

fT(W)

W

fT(W)

Fixed T


Idea of the proof



f TW

(W) : defined by power diagram

fT(W)

fixed WW


Idea of the proof



f TW(W) = minT fT(W)

fT(W)

fixed WW


Idea of the proof



TW(W) is concave !!

(because its graph is the lower

enveloppe of linear functions)

fT(W)

W

fT(W)



Idea of the proof



TW(W) is concave



fT(W)

W

fT(W)


Consider g(W) = f TWvj j


fT(W)

W

fT(W)


Consider g(W) = f TWvj j

vj - pow(yj)|| x yj ||2d (x) and g is concave. j =

Idea of the proof



TW(W) is concave



Part. 3 Optimal Transport the algorithm

Semi-discrete OT Summary:

Xc (x)d + Y (y)d

Sup

c(DMK) G( ) =



G( ) = g(W) = j Lag (yj) || x yj ||2 - (yj) d + j (yj) vj is concave

Xc (x)d + Y (y)d

Sup

c(DMK) G( ) =




vj - pow(yj)|| x yj ||2d (x) (= 0 at the maximum) G / j =

Xc (x)d + Y (y)d

Sup

c(DMK) G( ) =




vj - pow(yj)|| x yj ||2d (x) (= 0 at the maximum) G / j =

Xc (x)d + Y (y)d

Sup

c(DMK) G( ) =

Desired mass at yj Mass transported to yj


The [AHA] paper summary:

The optimal weights minimize a convex function

The gradient of this convex function is easy to compute

Note: the weight w(s) correspond to the Kantorovich potential (x)

Monge-

The algorithm:

Input: two tetrahedral meshes M1 and M2

Output: a morphing between M1 and M2






Monge-

The algorithm:



Step 1: sample M2 with N points (s1 sN)






Monge-

The algorithm:




Step 2: initialize the weights (w1 wN) = (0 0)






Monge-

The algorithm:




Step 2: initialize the weights (w1 wN

Step 3: minimize g(w1 wN) with a quasi-Newton algorithm:






Monge-

The algorithm:






For each iterate (s1 sN)(k):






Monge-

The algorithm:




Step 2: initialize the weights (w1 wN) = (0 0)



Compute Pow( (wi, si) M1 [Nivoliers, L 2014, Curves and Surfaces]


Compute Pow( (wi, si) M1 [Nivoliers, L 2014, Curves and Surfaces]

Implementation in GEOGRAM (http://alice.loria.fr/software/geogram






Monge-

The algorithm:







Compute Pow( (wi, si) M1 [L 2014, Curves and Surfaces]

Compute g and grad g






Monge-

The algorithm:







Compute Pow( (wi, si) M1 [L 2014, Curves and Surfaces]

Compute g and grad g

+ Multilevel version [Merigot 2011] (2D), [L 2014 arXiv, M2AN 2015] (3D)






Monge-

The algorithm:

Summary:

The algorithm computes the weights wi such that the power cells associated with

the Diracs correspond to the preimages of the Diracs.






Monge-

The algorithm:

Summary:








Monge-

The algorithm:

Summary:








Monge-

The algorithm:

Summary:








Monge-

The algorithm:

Summary:



Understanding Going

4

Part. 4 Optimal Transport ???

Wait a minute:

This means that one can move (and possibly deform)

a power diagram simply by changing the weights ?

Part. 4 Optimal Transport ???

Wait a minute:

This means that one can move (and possibly deform)

a power diagram simply by changing the weights ?

Reminder: Power diagram in 2d = intersection between

Voronoi diagram in 3d and IR2

hi = sqrt(wmax wi)

Height of point i Weight of point i

Part. 4 Power Diagrams & Transport







hi





Part. 4 Optimal Transport still lost ? 2D examples

Numerical Experiment: Simple translation


Part. 4 Power Diagrams and Transport

Part. 4 Optimal Transport 2D examples

Numerical Experiment: Splitting a disk


Numerical Experiment: A disk becomes two disks


Numerical Experiment: A sphere becomes a cube


Numerical Experiment: A sphere becomes two spheres


Numerical Experiment: Armadillo to sphere


Numerical Experiment: Other examples


Numerical Experiment: Varying density


Numerical Experiment: Performances



Note that a few years ago, several hours of supercomputer time were needed

for computing OT with a few thousand Dirac masses, with a combinatorial

algorithm in O(n3)





algorithm in O(n3)

With the semi-discrete algorithm, it takes less than 10 seconds on my laptop





algorithm in O(n3)

In the semi-discrete setting, my 3D version of multigrid algorithm

computes OT for 1 million Dirac masses in less than 1 hour on a laptop PC





algorithm in O(n3)

In the semi-discrete setting, my 3D version of multigrid algorithm

computes OT for 1 million Dirac masses in less than 1 hour on a laptop PC

Even much faster convergence can probably be reached with a true Newton

Conclusions Open questions

* Connections with physics, Legendre transform and entropy ?

[Cuturi & Peyré] regularized discrete optimal transport why does it work ?

Hint 1: Minimum action principle subject to conservation laws

* More continuous numerical algorithms ?

[Benamou & Brenier] fluid dynamics point of view very elegant, but 4D problem !!

FEM-type adaptive discretization of the subdifferential (graph of T) ?

* Can we characterize OT in other semi-discrete settings ?

measures supported on unions of spheres

piecewise linear densities

* Connections with computational geometry ?

Singularity set [Figalli] = set of points where T is discontinuous

Voronoi diagrams

Conclusions - ReferencesSome references (that this presentation is based on)

A Multiscale Approach to Optimal Transport,

Quentin Mérigot, Computer Graphics Forum, 2011

Variational Principles for Minkowski Type Problems, Discrete Optimal Transport,

and Discrete Monge-Ampere Equations

Xianfeng Gu, Feng Luo, Jian Sun, S.-T. Yau, ArXiv 2013

Minkowski-type theorems and least-squares clustering

AHA! (Aurenhammer, Hoffmann, and Aronov), SIAM J. on math. ana. 1998

Topics on Optimal Transportation, 2003

Optimal Transport Old and New, 2008

Cédric Villani

Conclusions - ReferencesOther references (that I need to read and understand)

Polar factorization and monotone rearrangement of vector-valued functions

Yann Brenier, Comm. On Pure and Applied Mathematics, June 1991

A computational fluid mechanics solution of the Monge-Kantorovich mass transfer

problem, J.-D. Benamou, Y. Brenier, Numer. Math. 84 (2000), pp. 375-393

Pogorelov, Alexandrov Gradient maps, Minkovsky problem (older than AHA

paper, some overlap, in slightly different context, formalism used by Gu & Yau)

Rockafeller Convex optimization Theorem to switch inf(sup()) sup(inf())

with convex functions (used to justify Kantorovich duality)

New textbook: Filippo Santambrogio Optimal Transport for Applied

Mathematician, Calculus of Variations, PDEs and Modeling Jan 15, 2015

Online resources

All the sourcecode/documentation available from:

alice.loria.fr/software/geogram

Computes semi-discrete OT in 3D

Scales up to millions Dirac masses on a laptop

L., A numerical algorithm for semi-discrete L2 OT in 3D,

ESAIM Math. Modeling and Analysis, accepted

(draft: http://arxiv.org/abs/1409.1279 <= to be fixed: bug

in MA equation in this version, fixed in M2AN journal version)

http://arxiv.org/abs/1409.1279

Acknowledgements

Funding: European Research Council french

GOODSHAPE ERC-StG-205693

VORPALINE ERC-PoC-334829

ANR MORPHO, ANR BECASIM

Quentin Merigot, Yann Brenier, Boris Thibert, Emmanuel Maitre,

Jean-David Benamou, Filippo Santambrogio, Edouard Oudet, Hervé Pajot.

ANR TOMMI, ANR GEOMETRYA

Science

Optimal Transport for a Computer Programmer's Point of View