  Nuclear Density Functional Theory (DFT) plays a prominent role in the
understanding of nuclear structure, being the approach with the widest range of
applications. Hohenberg and Kohn theorems warrant the existence of a nuclear
Energy Density Functional (EDF), yet its form is unknown. Current efforts to
build a nuclear EDF are hindered by the lack of a strategy for systematic
improvement. In this context, alternative approaches should be pursued and, so
far, an unexplored avenue is that related to the inverse DFT problem. DFT is
based on the one-to-one correspondence between Kohn-Sham (KS) potentials and
densities. The exact EDF produces the exact density, so that from the knowledge
of experimental or {\it ab initio} densities one may deduce useful information
through reverse engineering. The idea has already been proven to be useful in
the case of electronic systems. The general problem should be dealt with in
steps, and the objective of the present work is to focus on testing algorithms
to extract the Kohn-Sham potential within the simplest ansatz from the
knowledge of the experimental neutron and proton densities. We conclude that
while robust algorithms exist, the experimental densities present some critical
aspects. Finally, we provide some perspectives for future works.
