Reconstruction of the early Universe as a convex optimization problem

By Yann Brenier, U. Frisch, M. Hénon, Grégoire Loeper, S. Matarrese, Roya Mohayaee, A. Sobolevskiĭ. 2003

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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–Ampere–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–Ampere 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/f6a9bc14408aa478a4ee7139cfa4106248db225c

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Citations in the corpus

Improving Cosmological Distance Measurements by Reconstruction of the Baryon Acoustic Peak (2006, methodology): “The techniques of Croft & Gaztanaga (1997) and Brenier et al. (2003) are based on the principle of least action again, with the approximation that particles move on straight trajectories.”
The Maslov dequantization, idempotent and tropical mathematics: A brief introduction (2005, background): “idempotent functional analysis (e.g., the problem of eigenfunctions for “idempotent” integral operators, see [118]). Applications of optimal transport to data processing in cosmology are presented in [9,40]. 13. Relations to logic,fuzzy sets,and possibilitytheory Let S be an idempotent semiring with”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “By the way, Appendix A in [89] explains in detail the derivation of the above equations in co-moving coordinates presenting the set of choices and simplifications that have been made compared to the general framework in [45].”
Bayesian physical reconstruction of initial conditions from large-scale structure surveys (2012, background): “The field of velocity field reconstructions has a long history (see e.g. Bertschinger et al. 1990; Nusser & Dekel 1992; Dekel et al. 1999; Frisch et al. 2002; Brenier et al. 2003; Mohayaee & Sobolevskiĭ 2008; Lavaux 2008; Kitaura et al. 2011).”
Burgers Turbulence (2007, methodology): “Test of the MAK reconstruction for a sample of N ′ = 17, 178 points from a N-body simulation (from [25]).”
Evolution of the cosmic web (2014, methodology): “The application of the Monge–Ampère–Kantorovich reconstruction algorithm (Frisch et al. 2002; Brenier et al. 2003) to the characterization of the nature of voids (Lavaux & Wandelt 2010) is an interesting example.”