Authors: Bernd Hofmann and Otmar Scherzer
Abstract: We discuss interdependencies of the terms of a formal Taylor series expansion F(x+h)-F(x)=F'(x)h + R(x,h) in the context of a nonlinear ill-posed problem F(x)=y. Almost every convergence analysis based on Taylor series expansion uses the fact that the remainder R(x,h) becomes small for sufficiently small h. Since for compact operators F the linear part ||F'(x)h|| may be significantly small compared with the residual norm ||R(x,h)||, it seems to be important to characterize ||F(x+h)-F(x)-F'(x)h|| with respect to ||F(x+h)-F(x)||, ||F'(x)h|| and ||h|| for ill-posed problems. In this way definitions of a local degree of nonlinearity and of a local degree of ill-posedness have a common motivation. There are two extrema cases: ||F(x+h)-F(x)-F'(x)h|| =< q ||F(x+h)-F(x)||, where q < 1, and ||F(x+h)-F(x)-F'(x)h|| =< q ||h||^2. The later estimate applies to Frechet-differentiable operators F with a Lipschitz continuous derivative. In general this estimate does not guarantee a correlation between the nonlinear part F(x+h)-F(x) and its linearization F'(x)h. In contrast, the former estimate is associated with the case that the terms F'(x)h and F(x+h)-F(x) are closely related. For a wide class of nonlinearity degrees, Höolder rates for Tikhonov regularization are obtained under source conditions. Applications of the degree of nonlinearity are given.