  We investigate the superfluid-insulator transition of one-dimensional
interacting Bosons in both deep and shallow periodic potentials. We compare a
theoretical analysis based on Monte-Carlo simulations in continuum space and
Luttinger liquid approach with experiments on ultracold atoms with tunable
interactions and optical lattice depth. Experiments and theory are in excellent
agreement. It provides a quantitative determination of the critical parameter
for the Mott transition and defines the regime of validity of widely used
approximate models, namely the Bose-Hubbard and sine-Gordon models
