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2012

From Knothe's rearrangement to Brenier's optimal transport map

Bonnotte, Nicolas

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The Brenier optimal map and the Knothe-Rosenblatt rearrangement are two instances of a transport map, that is to say a map sending one measure onto another.

The main interest of the former is that it solves the Monge-Kantorovich optimal transport problem, while the latter is very easy to compute, being given by an explicit formula.
A few years ago, Carlier, Galichon, and Santambrogio showed that the Knothe rearrangement could be seen as the limit of the Brenier map when the quadratic cost degenerates.
In this paper, we prove that on the torus (to avoid boundary issues), when all the data are smooth, the evolution is also smooth, and is entirely determined by a PDE for the Kantorovich potential (which determines the map), with a subtle initial condition.

From Knothe's rearrangement to Brenier's optimal transport map · Around

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The original paper was published in Dokl. Akad. Nauk SSSR
L. Kantorovich · 1942
Earlier work this paper cites.
M. Rosenblatt
1952
Earlier work this paper cites.
H. Knothe
1957
Earlier work this paper cites.
D. Gilbarg and N. S. Trudinger
1977
Earlier work this paper cites.
R. S. Hamilton
1982
Earlier work this paper cites.

Similar

Y. Brenier
1987
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L. A. Caffarelli
1992
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D. Cordero-Erausquin
1999
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J.-D. Benamou and Y. Brenier
2000
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R. J. McCann
2001
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G. Monge
Cited in the paper.

Then

S. Angenent, S. Haker, and A. Tannenbaum
2003
Later among the works it cites.
G. Loeper and F. Rapetti
2005
Later among the works it cites.
G. Carlier, A. Galichon, and F. Santambrogio
2009
Later among the works it cites.
A. Figalli, F. Maggi, and A. Pratelli
2010
Later among the works it cites.
To appear in J. Eur. Math. Soc. (JEMS), 2011
A. Figalli, Y.-H. Kim, and R. J. McCann · 2011
Later among the works it cites.

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