A multi-level method for the determination of nonlinear coefficient functions of parabolic partial differential equations is presented as an example of the reconstruction of the hydraulic properties of porous media from soil column outflow experiments. A linear stability and error analysis is used to characterize the ill-posedness of the inverse problem. Founded on this analysis, optimal experimental design problems can be formulated. Numerical examples are presented which show that the method described leads to a stable solution of the inverse problem.