  Motion of a continuous fluid can be decomposed into an "incompressible"
rearrangement, which preserves the volume of each infinitesimal fluid element,
and a gradient map that transfers fluid elements in a way unaffected by any
pressure or elasticity (the polar decomposition of Y. Brenier). The Euler
equation describes a system whose kinematics is dominated by the incompressible
rearrangement. The opposite limit, in which the incompressible component is
negligible, corresponds to the Zel'dovich approximation, a model of motion of
self-gravitating fluid in cosmology.
  We present a method of approximate reconstruction of the large-scale proper
motions of matter in the Universe from the present-day mass density field. The
method is based on recovering the corresponding gradient transfer map. We
discuss its algorithmics, tests of the method against mock cosmological
catalogues, and its application to observational data, which result in tight
constraints on the mean mass density Omega_m and age of the Universe.
