On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces

We consider generalized inverses and linear ill-posed problems in Banach spaces, and the concept of pseudo-optimal parameter choices and stopping rules for regularization methods is presented. The pseudo-optimality of the discrepancy principle for iterative methods like the Richardson iteration is shown, as well as the pseudo-optimality of different parameter choices for the iterated method of Lavrentiev. AMS (MOS) Subject Classification (1991): 65J20, 65R30. 1 Introduction It is our purpose to present pseudo-optimal parameter choices and stopping rules for regularization methods with respect to various norms, e.g., L p -norms or the maximum norm, and this extend certain known results in Hilbert spaces. Another purpose is to consider only those regularization methods that avoid any transformation (in Hilbert spaces that means the transformation to normal equations, for example) of the underlying problem in order to keep computational costs as well as condition numbers at the lowest ..