  We benchmark the ground state energies and the density profiles of atomic
repulsive Fermi gases in optical lattices computed via Density Functional
Theory (DFT) against the results of diffusion Monte Carlo (DMC) simulations.
The main focus is on a half-filled one-dimensional optical lattices, for which
the DMC simulations performed within the fixed-node approach provide unbiased
results. This allows us to demonstrate that the local spin-density
approximation (LSDA) to the exchange-correlation functional of DFT is very
accurate in the weak and intermediate interactions regime, and also to
underline its limitations close to the strongly-interacting Tonks-Girardeau
limit and in very deep optical lattices. We also consider a three dimensional
optical lattice at quarter filling, showing also in this case the high accuracy
of the LSDA in the moderate interaction regime. The one-dimensional data
provided in this study may represent a useful benchmark to further develop DFT
methods beyond the LSDA and they will hopefully motivate experimental studies
to accurately measure the equation of state of Fermi gases in
higher-dimensional geometries.
