Mathematics – Logic
Scientific paper
Apr 2003
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003eaeja....14884f&link_type=abstract
EGS - AGU - EUG Joint Assembly, Abstracts from the meeting held in Nice, France, 6 - 11 April 2003, abstract #14884
Mathematics
Logic
Scientific paper
The Monge-Kantorovich mass transportation problem dates back to work by Monge in 1781 on how to optimally move earth from one place to another, knowing only the initial and final landscapes, the cost being a prescribed function of the distance travelled by "molecules" of earth. We solve the cosmological reconstruction problem of mapping the present locations of (mostly dark) matter, to their primordial locations, knowing only the current field of mass density, e.g. from a full-sky galaxy catalogue or a numerical simulation. Under the assumption that the map is close to potential, we reduce the problem to solving a nonlinear partial differential equation, originally written by Ampere in 1820, now known as the Monge-Ampere equation. Thanks to recent work by Y. Brenier, this becomes a Monge-Kantorovich problem with quadratic cost function and, in discretised form, an assignment problem: find the pairing between N departure and N arrival points which minimises the sum of the squared distances between paired points. The latter can be solved very efficiently by the auction algorithm of Bertsekas. When tested against N-body cosmological simulations, excellent reconstruction is obtained above a few megaparsecs. Based on the paper Frisch-Matarrese-Mohayaee-Sobolevski Nature 417, 260-262 (16 May 2002).
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