A fast semidiscrete optimal transport algorithm for a unique reconstruction of the early Universe

Abstract

We leverage powerful mathematical tools stemming from optimal transport theory and transform them into an efficient algorithm to reconstruct the fluctuations of the primordial density field, built on solving the Monge-Ampère-Kantorovich equation. Our algorithm computes the optimal transport between an initial uniform continuous density field, partitioned into Laguerre cells, and a final input set of discrete point masses, linking the early to the late Universe. While existing early universe reconstruction algorithms based on fully discrete combinatorial methods are limited to a few hundred thousand points, our algorithm scales up well beyond this limit, since it takes the form of a well-posed smooth convex optimization problem, solved using a Newton method. We run our algorithm on cosmological N-body simulations, from the AbacusCosmos suite, and reconstruct the initial positions of $\mathcal {O}(10^7)$ particles within a few hours with an off-the-shelf personal computer. We show that our method allows a unique, fast and precise recovery of subtle features of the initial power spectrum, such as the baryonic acoustic oscillations.

Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries

“…where ν : Ω → R + is a square-integrable density 4 . In the constraint (second line), ψ c corresponds to the Legendre-Fenchel transform of ψ, and the condition that applying it twice to ψ does not change ψ means that ψ is convex (because the graph of ψ cc corresponds to the convex hull of the graph of ψ).…”

“…• second, the ghost cells have a significant computational cost: computing a Laguerre diagram costs O(N d √ N ) (where d ∈ {2, 3} is the dimension), and the Newton algorithm converges in O(N log(N )) iterations (empirical results in [4]). The "ghost cells" techniques computes a diagram with N + M vertices (instead of N ), where M needs to be sufficiently large to accurately capture the free boundary.…”

“…[1]), the volume of fluid is conserved, while the shape of the fluid can considerably vary throughout the simulation, and can change topology (split and merge). In some astrophysic simulations [2,3,4], the Universe considered at a cosmological scale can be modeled as a "fluid" 1 . With the simplifying assumption that this fluid obeys the incompressible Euler equation, one can reconstruct the full dynamics of the Universe from the knowledge of the density fluid at current time [2,3,4].…”

“…In some astrophysic simulations [2,3,4], the Universe considered at a cosmological scale can be modeled as a "fluid" 1 . With the simplifying assumption that this fluid obeys the incompressible Euler equation, one can reconstruct the full dynamics of the Universe from the knowledge of the density fluid at current time [2,3,4]. To name another example, in shape and 1 The "atoms" of this "fluid" correspond to galaxy clusters !…”

“…The approach builds on recent advances in numerical optimal transport, that resulted in Lagrangian schemes for fluid simulation [7] or early Universe reconstruction [4]. In the works cited above, the object fills the entire simulation domain.…”