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 parallel-to F'(x)h parallel-to may be significantly small compared with the residual norm parallel-to R(x, h) parallel-to, it seems to be important to characterize parallel-to F(x + h) - F(x) - F'(x)h parallel-to with respect to parallel-to F(x + h) - F(x) parallel-to, parallel-to F'(x)h parallel-to and parallel-to h parallel-to for ill-posed problems. In this way, definitions of a local degree of nonlinearity and of a local degress of ill-posedness have a common motivation. There are two extreme cases: parallel-to F(x + h) - F(x) - F'(x)h parallel-to less-than-or-equal-to q parallel-to q parallel-to F(x + h) - F(x) parallel-to, where q < 1, and parallel-to F(x + h ) - F(x) - F'(x)h parallel-to less-than-or-equal-to q parallel-to h parallel-to2. The later 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.