LAUSR

A weighted, semidiscrete, fast optimal transport (OT) algorithm for reconstructing the Lagrangian positions of protohalos from their evolved Eulerian positions is presented. The algorithm makes use of a mass estimate of the biased tracers and of the distribution of the remaining mass (the "dust") but is robust to errors in the mass estimates. Tests with state-of-art cosmological simulations show that if the dust is assumed to have a uniform spatial distribution, then the shape of the OT-reconstructed pair correlation function of the tracers is very close to linear theory, enabling subpercent precision in the baryon acoustic oscillation distance scale that depends weakly, if at all, on a cosmological model. With a more sophisticated model for the dust, OT returns an estimate of the displacement field which yields superb reconstruction of the protohalo positions and, hence, of the shape and amplitude of the initial pair correlation function of the tracers. This enables direct and independent determinations of the bias factor b and the smearing scale Σ, potentially providing new methods for breaking the degeneracy between b and σ_{8}.