Andrei Nikolaevich Kolmogorov - OCA
Transcript of Andrei Nikolaevich Kolmogorov - OCA
Andrei Nikolaevich Kolmogorov
KOLMOGOROV 100 – p.1/17
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
Brief history of Universe
NASA−GODDARDWMAP Feb. 2003
present massfield
past history(unique?)
Peebles (1989)
KOLMOGOROV 100 – p.3/17
History of Universe reconstructed
NASA−GODDARDWMAP Feb. 2003
present massfield
past history(unique?)
Peebles (1989)
KOLMOGOROV 100 – p.3/17
Fluid dynamics in expanding universe
Euler:�� � � ��� �
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Mass conservation:
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Poisson:
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KOLMOGOROV 100 – p.4/17
Fluid dynamics in expanding universe
Euler:�� � � ��� �
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Mass conservation:
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Poisson:
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KOLMOGOROV 100 – p.4/17
Fluid dynamics in expanding universe
Euler:�� � � ��� �
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Mass conservation:
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Poisson:
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Zeldovich approximation:
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late 1960s
KOLMOGOROV 100 – p.4/17
Potential Lagrangian map“Zeldovich” equation:
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. . . (graph) invertible if is convex
Bertschinger–Dekel (1989)
KOLMOGOROV 100 – p.5/17
Potential Lagrangian map“Zeldovich” equation:
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. . . (graph) invertible if is convex
Bertschinger–Dekel (1989)
KOLMOGOROV 100 – p.5/17
Potential Lagrangian map“Zeldovich”/Burgers equation:
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. . . (graph) invertible if is convex
Bertschinger–Dekel (1989)
KOLMOGOROV 100 – p.5/17
Potential Lagrangian map“Zeldovich”/Burgers equation:
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Bertschinger–Dekel (1989)KOLMOGOROV 100 – p.5/17
Potential Lagrangian map“Zeldovich”/Burgers equation:
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. . . (graph) invertible if is convex
Bertschinger–Dekel (1989)KOLMOGOROV 100 – p.5/17
The Monge–Ampère equation
Mass conservation �
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KOLMOGOROV 100 – p.6/17
The Monge–Ampère equation
Mass conservation, �� �
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KOLMOGOROV 100 – p.6/17
The Monge–Ampère equation
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KOLMOGOROV 100 – p.6/17
The Monge–Ampère equation
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KOLMOGOROV 100 – p.6/17
The Monge–Ampère equation
Mass conservation, �� �
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prescribed
1820
KOLMOGOROV 100 – p.6/17
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
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
Monge’s mass transportation problem
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�minimize
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over all
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such that �� ��
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KOLMOGOROV 100 – p.9/17
Monge, Ampère, mass transportation
Theorem (Brenier 1987, 1991) The minimizing mapsare gradients of convex functions:
and solve suitable Monge–Ampere equations
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�minimize
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over all
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such that �� ��
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KOLMOGOROV 100 – p.10/17
Monge, Ampère, mass transportation
Theorem (Brenier 1987, 1991) The minimizing mapsare gradients of convex functions:
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and solve suitable Monge–Ampere equations
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�minimize
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over all
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such that �� ��
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KOLMOGOROV 100 – p.10/17
Kantorovich relaxation
For given �� ��
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minimize
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over all � ��
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such that
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1942
KOLMOGOROV 100 – p.11/17
Discretization and assignment
Discrete densities:
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Minimize the cost���� �
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over all permutations�� � � �
of
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τ=0
τ=τ0
KOLMOGOROV 100 – p.12/17
Relaxation and the dual problem
Minimize
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over all permutations
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Dual problem:
Minimize
Hénon 1950s–1990s
KOLMOGOROV 100 – p.13/17
Relaxation and the dual problem
Minimize
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over all bistochastic matrices:
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Dual problem:
Minimize
Hénon 1950s–1990s
KOLMOGOROV 100 – p.13/17
Relaxation and the dual problem
Minimize
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over all bistochastic matrices:
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Dual problem:
Minimize
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Hénon 1950s–1990s
KOLMOGOROV 100 – p.13/17
Relaxation and the dual problem
Minimize
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over all bistochastic matrices:
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Dual problem:
Minimize
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Hénon 1950s–1990s
KOLMOGOROV 100 – p.13/17
Time complexity
� Burkard & Derigs 1980
Bertsekas 1979–2003
“dense” auction
� “sparse” auction
(time in secondsdivided by
�
)10310
KOLMOGOROV 100 – p.14/17
Testing against -body simulation
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of� � � ��� �
Mpc size(middle 10% slice)
KOLMOGOROV 100 – p.15/17
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
KOLMOGOROV 100 – p.16/17
Euler–Poisson variational problem
Minimize
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� Mass conservation:
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Theorem (Loeper 2003) Up to a change of variables
this is a convex minimization problem with a unique solution
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KOLMOGOROV 100 – p.17/17