  We discuss the construction of a nuclear Energy Density Functional (EDF) from
ab initio calculations, and we advocate the need of a methodical approach that
is free from ad hoc assumptions. The equations of state (EoS) of symmetric
nuclear and pure neutron matter are computed using the chiral NNLO$_{\rm sat}$
and the phenomenological AV4$^\prime$+UIX$_{c}$ Hamiltonians as inputs in the
Self-consistent Green's Function (SCGF) and Auxiliary Field Diffusion Monte
Carlo (AFDMC) methods, respectively. We propose a convenient parametrization of
the EoS as a function of the Fermi momentum and fit it on the SCGF and AFDMC
calculations. We apply the ab initio-based EDF to carry out an analysis of the
binding energies and charge radii of different nuclei in the local density
approximation. The NNLO$_{\rm sat}$-based EDF produces encouraging results,
whereas the AV4$^\prime$+UIX$_{c}$-based one is farther from experiment.
Possible explanations of these different behaviors are suggested, and the
importance of gradient and spin-orbit terms is analyzed. Our work paves the way
for a practical and systematic way to merge ab initio nuclear theory and DFT,
while at the same time it sheds light on some of the critical aspects of this
procedure.
