A fast semi-discrete optimal transport algorithm for a unique
  reconstruction of the early Universe
  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-Amp\`ere-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.
