Well posedness and convergence of some regularisation methods for non-linear ill posed problems

The authors carry out a theoretical analysis of two regularisation methods for non-linear ill posed problems. The first is a penalty method called Tikhonov regularisation, in which one solves an unconstrained optimisation problem, while the second is based on a constrained optimisation problem. For each method they examine the well posedness of the respective optimisation problem. They then show strong convergence of the regularised 'solutions' to the true solution. (Note that this is well known for the application of these methods to linear problems.) They consider such factors as the convergence of perturbed data to the true data, inexact solution of the respective optimisation problems, and the choice of the regularisation parameters.