Factors influencing the ill-posedness of nonlinear problems

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)//, 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 extreme cases: //F(x+h)-F(x)-F'(x)h//<or=q//F(x+h)-F(x)//, where q<1, and //F(x+h)-F(x)-F(x)h//<or=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 where the terms F(x)h and F(x+h)-F(x) are closely related. For a wide class of nonlinearity degrees, Holder rates for Tikhonov regularization are obtained under source conditions. Applications of the degree of nonlinearity are given.