  In this work we investigate the nonlinear and nonlocal relation between
cosmological density and peculiar velocity fields. Our goal is to provide an
algorithm for the recon- struction of the nonlinear velocity field from the
fully nonlinear density. We find that including the gravitational tidal field
tensor using second order Lagrangian perturba- tion theory (2LPT) based upon an
estimate of the linear component of the nonlinear density field significantly
improves the estimate of the cosmic flow in comparison to linear theory not
only in the low density, but also and more dramatically in the high density
regions. In particular we test two estimates of the linear component: the log-
normal model and the iterative Lagrangian linearisation. The present approach
relies on a rigorous higher order Lagrangian perturbation theory analysis which
incorpo- rates a nonlocal relation. It does not require additional fitting from
simulations being in this sense parameter free, it is independent of
statistical-geometrical optimisation and it is straightforward and efficient to
compute. The method is demonstrated to yield an unbiased estimator of the
velocity field on scales ~> 5 Mpc/h with closely Gaussian distributed errors.
Moreover, the statistics of the divergence of the peculiar velocity field is
extremely well recovered showing a good agreement with the true one from N-body
simulations. The typical errors of about 10 km/s (1 sigma confidence intervals)
are reduced by more than 80% with respect to linear theory in the scale range
between 5 and 10 Mpc/h in high density regions ({\delta} > 2). We also find
that iterative Lagrangian linearisation is significantly superior in the low
density regime with respect to the lognormal model.
