  Generating mocks for future sky surveys requires large volumes and high
resolutions, which is computationally expensive even for fast simulations. In
this work we try to develop numerical schemes to calibrate various halo and
matter statistics in fast low resolution simulations compared to high
resolution N-body and hydrodynamic simulations. For the halos, we improve the
initial condition accuracy and develop a halo finder "relaxed-FOF", where we
allow different linking length for different halo mass and velocity
dispersions. We show that our relaxed-FoF halo finder improves the common
statistics, such as halo bias, halo mass function, halo auto power spectrum in
real space and in redshift space, cross correlation coefficient with the
reference halo catalog, and halo-matter cross power spectrum. We also
incorporate the potential gradient descent (PGD) method into fast simulations
to improve the matter distribution at nonlinear scale. By building a lightcone
output, we show that the PGD method significantly improves the weak lensing
convergence tomographic power spectrum. With these improvements FastPM is
comparable to the high resolution full N-body simulation of the same mass
resolution, with two orders of magnitude fewer time steps. These techniques can
be used to improve the halo and matter statistics of FastPM simulations for
mock catalogs of future surveys such as DESI and LSST.
