Ill-posedness and local ill-posedness concepts in hilbert spaces^*

^*Research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant Ho 1454/3-2

Abstract

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed

Keywords:

• Nonlinear Inverse Problems
• Ill-Posedness
• Stability Estimates
• Frechet Derivative
• Compact Operators
• Parameter Identification

^*Research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant Ho 1454/3-2

^*Research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant Ho 1454/3-2

Notes

^*Research was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant Ho 1454/3-2