  In previous works we proposed the Reverse Zeldovich Approximation (RZA)
method, which can be used to estimate the cosmological initial conditions
underlying the galaxy distribution in the Local Universe using peculiar
velocity data. In this paper, we apply the technique to run constrained
cosmological simulations from the RZA-reconstructed initial conditions,
designed to reproduce the large-scale structure of the Local Universe. We test
the method with mock peculiar velocity catalogues extracted from a reference
simulation. We first reconstruct the initial conditions of this reference
simulation using the mock data, and then run the reconstructed initial
conditions forward in time until z=0. We compare the resulting constrained
simulations with the original simulation at z=0 to test the accuracy of this
method. We also compare them with constrained simulations run from the mock
data without the addition of RZA, i.e. using only the currently established
constrained realizations (CR) method. Our re-simulations are able to correctly
recover the evolution of the large-scale structure underlying the data. The
results show that the addition of RZA to the CR method significantly improves
both the reconstruction of the initial conditions and the accuracy of the
obtained constrained resimulations. Haloes from the original simulation are
recovered in the re-simulations with an average accuracy of about 2 Mpc/h on
their position and a factor of 2 in mass, down to haloes with a mass of approx
10^14 M_odot/h. In comparison, without RZA the re-simulations recover only the
most massive haloes with masses of about 5x10^14 M_odot/h and higher, and with
a systematic shift on their position of about approx 10 Mpc/h due to the cosmic
displacement field. We show that with the additional Lagrangian reconstruction
step introduced by the RZA, this shift can be removed.
