Reconstruction of the early Universe as a convex optimization problem

Y. Brenier, U. Frisch, M. Hénon, G. Loeper, S. Matarrese, R. Mohayaee, A. Sobolevskiǐ

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93 Citations (Scopus)


We show that the deterministic past history of the Universe can be uniquely reconstructed from knowledge of the present mass density field, the latter being inferred from the three-dimensional distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales - a few Mpc - where multistreaming becomes significant. Above 6 h-1 Mpc we propose and implement an effective Monge-Ampère-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge. After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than O(N3) time complexity and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60 per cent of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampère equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. A self-contained discussion of relevant notions and techniques is provided.

Original languageEnglish
Pages (from-to)501-524
Number of pages24
JournalMonthly Notices of the Royal Astronomical Society
Issue number2
Publication statusPublished - 1 Dec 2003
Externally publishedYes


  • Cosmology: theory
  • Early Universe
  • Hydrodynamics
  • Large-scale structure of Universe

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