Regularization techniques in model fitting and parameter estimation

We consider fitting data by linear and nonlinear models. The specific problems that we aim at, although they encompass classic formulations, have as common ground the fact that we attack a special situation: the ill-posed problems. In the linear case, we consider the total least squares problem. There exist special methods to approach the so-called nongeneric cases, but we propose extensions for the more commonly encountered close-to-nongeneric problems. Several methods of introducing regularization in the context of total least squares are analyzed. They are based on truncation methods or on penalty optimization. The obtained problems might not have closed form solutions. We discuss numerical linear algebra and local optimization methods. Data fitting by nonlinear or nonparametric models is the second subject of the thesis. We extend the nonlinear regression theory to the case when we have to deal with supplementary regularization constraints, and to a semiparametric context, where only part of the model is known and we have to take into account a component with unknown formulation. We apply the developed theory to the biomedical application of quantifying metabolite concentrations in the human brain from nuclear magnetic resonance spectroscopic signals.