Regularization of Linear Inverse Problems

The discretization of linear identification problems led to linear systems of algebraic equations. In case a solution does not exist, these can be replaced by linear least squares problems, as already done in Sect. 2.3 We will give a detailed sensitivity analysis of linear least squares problem and introduce their condition number as a quantitative measure of ill-posedness. If the condition of a problem is too bad, it can not be solved practically. We introduce and discuss the concept of regularization which formally consists in replacing a badly conditioned problem by a better conditioned one. The latter can be solved reliably, but whether its solution is of any relevance depends on how the replacement problem was constructed. We will especially consider Tikhonov regularization and iterative regularization methods. The following discussion is restricted to the Euclidean space over the field of real numbers \(\mathbb{K} = \mathbb{R}\), but could easily be extended to complex spaces.