Soft processes at the LHC, I: Multi-component model

We emphasize the sizeable effects of absorption on high-energy `soft' processes, and, hence, the necessity to include multi-Pomeron-Pomeron interactions in the usual multi-channel eikonal description. We present a model which includes a complete set of the multi-Pomeron vertices and which accounts for the diffusion in both, the impact parameter and ln(k_t), of the parton during its evolution in rapidity. We tune the model to the available data for soft processes in the CERN-ISR to Tevatron energy range. We make predictions for `soft' observables at the LHC.