2006, Mathé, Peter, Hofmann, Bernd

The stable approximate solution of ill-posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise-free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise-free part which decrease to zero along with the regularization parameter are called profile functions and are subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of source-wise representations is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces ...

2010, Flemming, Jens, Hofmann, Bernd, Mathé, Peter

In the analysis of ill-posed inverse problems the impact of solution smoothness on accuracy and convergence rates plays an important role. For linear ill-posed operator equations in Hilbert spaces and with focus on the linear regularization schema we will establish relations between the different kinds of measuring solution smoothness in a point-wise or integral manner. In particular we discuss the interplay of distribution functions, profile functions that express the regularization error, index functions generating source conditions, and distance functions associated with benchmark source conditions. We show that typically the distance functions and the profile functions carry the same information as the distribution functions, and that this is not the case for general source conditions. The theoretical findings are accompanied with examples exhibiting applications and limitations of the approach