Andrei Nikolaevich Kolmogorov - OCA

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Andrei Nikolaevich Kolmogorov KOLMOGOROV 100 – p.1/17

Transcript of Andrei Nikolaevich Kolmogorov - OCA

Page 1: Andrei Nikolaevich Kolmogorov - OCA

Andrei Nikolaevich Kolmogorov

KOLMOGOROV 100 – p.1/17

Page 2: Andrei Nikolaevich Kolmogorov - OCA

Back to the primordial universeby a Monge–Ampère–Kantorovich

mass transportation methodUriel FRISCH

Observatoire de la Cote d’Azur, Nice, France

� Y. Brenier, U. Frisch, M. Henon, G. Loeper, S. Matarrese, R. Mohayaee,

A. Sobolevskiı Mon. Not. R. Astron. Soc. (2003, submitted), astro-ph/0304214

� U. Frisch, S. Matarrese, R. Mohayaee, A. Sobolevski Nature 417 (2002) 260–262

KOLMOGOROV 100 – p.2/17

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Brief history of Universe

NASA−GODDARDWMAP Feb. 2003

present massfield

past history(unique?)

Peebles (1989)

KOLMOGOROV 100 – p.3/17

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History of Universe reconstructed

NASA−GODDARDWMAP Feb. 2003

present massfield

past history(unique?)

Peebles (1989)

KOLMOGOROV 100 – p.3/17

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Fluid dynamics in expanding universe

Euler:�� � � ��� �

� � �

� �� � � ����

Mass conservation:

�� � ��

� � � � � �

Poisson:

�� ��� �

� ��

KOLMOGOROV 100 – p.4/17

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Fluid dynamics in expanding universe

Euler:�� � � ��� �

� � �

� �� � � ����

Mass conservation:

�� � ��

� � � � � �

Poisson:

�� ��� �

� ��

“Slaving” as � �:

��� �

� � �� � ��

� �

�� �

� �

KOLMOGOROV 100 – p.4/17

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Fluid dynamics in expanding universe

Euler:�� � � ��� �

� � �

� �� � � ����

Mass conservation:

�� � ��

� � � � � �

Poisson:

�� ��� �

� ��

Zeldovich approximation:

� � � ��

late 1960s

KOLMOGOROV 100 – p.4/17

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Potential Lagrangian map“Zeldovich” equation:

�� � �� � � � �

. . . (graph) invertible if is convex

Bertschinger–Dekel (1989)

KOLMOGOROV 100 – p.5/17

Page 9: Andrei Nikolaevich Kolmogorov - OCA

Potential Lagrangian map“Zeldovich” equation:

�� � �� � � � �

��� �

� � �� � ��

� � � � ���

. . . (graph) invertible if is convex

Bertschinger–Dekel (1989)

KOLMOGOROV 100 – p.5/17

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Potential Lagrangian map“Zeldovich”/Burgers equation:

�� � �� � � � �

��� �

� � �� � ��

� � � � ���

. . . (graph) invertible if is convex

Bertschinger–Dekel (1989)

KOLMOGOROV 100 – p.5/17

Page 11: Andrei Nikolaevich Kolmogorov - OCA

Potential Lagrangian map“Zeldovich”/Burgers equation:

�� � �� � � � �

��� �

� � �� � ��

� � � � ���

� � � � �� ��

�� � � � �

� � ��

��

�� �

� � � � � ��

��

� ��

. . . (graph) invertible if is convex

Bertschinger–Dekel (1989)KOLMOGOROV 100 – p.5/17

Page 12: Andrei Nikolaevich Kolmogorov - OCA

Potential Lagrangian map“Zeldovich”/Burgers equation:

�� � �� � � � �

��� �

� � �� � ��

� � � � ���

� � � � �� ��

�� � � � �

� � ��

��

�� �

� � � � � ��

��

� ��

. . . (graph) invertible if is convex

Bertschinger–Dekel (1989)KOLMOGOROV 100 – p.5/17

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The Monge–Ampère equation

Mass conservation �

��� �

� � �

�� ��

��

��� ��

��

� � � initial densitypresent density

KOLMOGOROV 100 – p.6/17

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The Monge–Ampère equation

Mass conservation, �� �

� � �

��� �

� � �

��� ��

��

� � � initial densitypresent density

KOLMOGOROV 100 – p.6/17

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The Monge–Ampère equation

Mass conservation, �� �

� � �

��� �

� � �

��� ��

��

� � � initial densitypresent density

As ��

�� � �

��

� �

��� � ���� ���

��

� � �

����

��

� �

KOLMOGOROV 100 – p.6/17

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The Monge–Ampère equation

Mass conservation, �� �

� � �

��� �

� � �

��� ��

��

� � � initial densitypresent density

As ��

�� � �

��

� �

��� � ���� ���

��

� � �

����

��

� �

Legendre–Fenchel transform �

��

� � � ���

��� � �

�� �

KOLMOGOROV 100 – p.6/17

Page 17: Andrei Nikolaevich Kolmogorov - OCA

The Monge–Ampère equation

Mass conservation, �� �

� � �

��� � � � ��� ��

��

As ��

�� � ��

�� �

��� � �� � � �

��

� � � ��� ��

��� ��

prescribed

1820

KOLMOGOROV 100 – p.6/17

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Monge’s mass transportation problem

déblai remblai. . .Il n’est pas indifferent que telle molecule de deblai soit transportee

dans tel ou tel autre endroit du remblai, mais qu’il y a une certaine

distribution a faire des molecules du premier dans le second, d’apres

laquelle la somme de ces produits sera la moindre possible, et le prix du

transport total sera un minimum.

KOLMOGOROV 100 – p.7/17

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Monge’s mass transportation problem

cut fill. . .It is not indifferent that any given molecule of the cuts be transported

to this or that place in the fills, but there ought to be a certain distribution

of molecules of the former into the latter, according to which the sum of

these products will be the least possible, and the cost of transportation

will be a minimum.

KOLMOGOROV 100 – p.8/17

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Monge’s mass transportation problem

� �

For given �� ��

��

, ��� ��

�minimize

��

��

� �� �� ��

�� � � � �

� ��

�� � �� �

�� � �

over all

��

��

��

��

�� �

such that �� ��

�� � � � ��� �

�� � �

KOLMOGOROV 100 – p.9/17

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Monge, Ampère, mass transportation

Theorem (Brenier 1987, 1991) The minimizing mapsare gradients of convex functions:

and solve suitable Monge–Ampere equations

For given �� ��

��

, ��� ��

�minimize

��

��

� �� � �� ��

�� � � � �

� ��

�� � � �� �

�� � �

over all

��

��

��

��

�� �

such that �� ��

�� � � � ��� �

�� � �

KOLMOGOROV 100 – p.10/17

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Monge, Ampère, mass transportation

Theorem (Brenier 1987, 1991) The minimizing mapsare gradients of convex functions:

��

�� � �

��

��

��

�� � ��

��

and solve suitable Monge–Ampere equations

For given �� ��

��

, ��� ��

�minimize

��

��

� �� � �� ��

�� � � � �

� ��

�� � � �� �

�� � �

over all

��

��

��

��

�� �

such that �� ��

�� � � � ��� �

�� � �

KOLMOGOROV 100 – p.10/17

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Kantorovich relaxation

For given �� ��

��

, ��� ��

minimize

�� �

� � � ��

�� � � � �

over all � ��

��

such that

� ��

�� � � � �� ��

��

� ��

�� � � � ��� �

��

1942

KOLMOGOROV 100 – p.11/17

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Discretization and assignment

Discrete densities:

��� ��

� ��

��� �� �

� � ���

�� ��

�� �

�� � �

� �� � ��

Minimize the cost���� �

�� � � �� ��

� �

over all permutations�� � � �

of

� ��

��� � � �

τ=0

τ=τ0

KOLMOGOROV 100 – p.12/17

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Relaxation and the dual problem

Minimize

�� � �

�� � � �� ��

� �

over all permutations

�� � � �

Dual problem:

Minimize

Hénon 1950s–1990s

KOLMOGOROV 100 – p.13/17

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Relaxation and the dual problem

Minimize

���� � � �

�� � � �� � � �

over all bistochastic matrices:

� � ��

�� � �

� � ��

� � �

�� � �

Dual problem:

Minimize

Hénon 1950s–1990s

KOLMOGOROV 100 – p.13/17

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Relaxation and the dual problem

Minimize

���� � � �

�� � � �� � � �

over all bistochastic matrices:

� � ��

�� � �

� � ��

� � �

�� � �

Dual problem:

Minimize

�� � �

� � �

� � �

� � � �� � � �� �

Hénon 1950s–1990s

KOLMOGOROV 100 – p.13/17

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Relaxation and the dual problem

Minimize

���� � � �

�� � � �� � � �

over all bistochastic matrices:

� � ��

�� � �

� � ��

� � �

�� � �

Dual problem:

Minimize

�� � �

� � �

� � �

� � � �� � � �� �

Hénon 1950s–1990s

KOLMOGOROV 100 – p.13/17

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Time complexity

� Burkard & Derigs 1980

Bertsekas 1979–2003

“dense” auction

� “sparse” auction

(time in secondsdivided by

)10310

KOLMOGOROV 100 – p.14/17

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Testing against -body simulation

� �� �points in a box

of� � � ��� �

Mpc size(middle 10% slice)

KOLMOGOROV 100 – p.15/17

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Testing against -body simulation

0.20.2 0.3 0.7 0.80.4 0.5 0.6

0.4

0.3

0.5

0.7

0.8

0.6

simulation coordinate

reco

nstr

uctio

n co

ordi

nate

1 2 3 4distances

10

20

30

40

50

60

%

KOLMOGOROV 100 – p.15/17

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KOLMOGOROV 100 – p.16/17

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Euler–Poisson variational problem

Minimize

���

� � � �

� � � � � � � � � � �

� ���� �

� Mass conservation:

�� � � �

� � � � � �

� Poisson equation: �� �� �

� ��

� � ��

� � � �� ��

��

, � ��

� � � � �� ��

Theorem (Loeper 2003) Up to a change of variables

this is a convex minimization problem with a unique solution

� ��

��

���

.

KOLMOGOROV 100 – p.17/17