  Unitarization schemes can be reduced to non-linear equations which saturate
at small $b$ the evolution of the elastic amplitude with $s$, and which mimic
parton saturation in the non-perturbative regime. These equations enable us to
study the effect of saturation on total and elastic cross sections for various
models, and to evaluate the uncertainties on $\sigma_{tot}$ and $\rho(s,t)$ at
the LHC in the presence of a hard pomeron.
