A parameter estimation problem and its regularization by the conjugate gradient method

A parameter estimation problem with noisy data is considered which arises as an inverse problem in groundwater filtration. It turns out that this problem can be formulated as a linear non-compact ill-posed problem in appropriate Hilbert spaces, with an operator perturbation that can be estimated only at the solution of the problem.

Those problems are treated in a general setting and are numerically solved by the CGNR method, this is, the classical conjugate gradient method by Hestenes and Stiefel applied to the associated normal equations.

An a posteriori stopping rules is introduced to obtain stable numerical solutions, and convergence results are provided for the corresponding approximations.

Finally, numerical illustrations with the CGNR method applied to a non-compact linear perturbed test problem are provided.