  On the smallest scales, three-dimensional large-scale structure surveys
contain a wealth of cosmological information which cannot be trivially
extracted due to the non-linear dynamical evolution of the density field.
Lagrangian perturbation theory (LPT) is widely applied to the generation of
mock halo catalogs and data analysis. In this work, we compare topological
features of the cosmic web such as voids, sheets, filaments and clusters, in
the density fields predicted by LPT and full numerical simulation of
gravitational large-scale structure formation. We propose a method designed to
improve the correspondence between these density fields, in the mildly
non-linear regime. We develop a computationally fast and flexible tool for a
variety of cosmological applications. Our method is based on a remapping of the
approximately-evolved density field, using information extracted from N-body
simulations. The remapping procedure consists of replacing the one-point
distribution of the density contrast by one which better accounts for the full
gravitational dynamics. As a result, we obtain a physically more pertinent
density field on a point-by-point basis, while also improving higher-order
statistics predicted by LPT. We quantify the approximation error in the power
spectrum and in the bispectrum as a function of scale and redshift. Our
remapping procedure improves one-, two- and three-point statistics at scales
down to 8 Mpc/h.
