On the degree of ill-posedness for linear problems with noncompact operators. (English) Zbl 1228.47017

Hofmann, Bernd; Kindermann, Stefan

Methods Appl. Anal. 17, No. 4, 445-462 (2010).

Authors’ abstract: In inverse problems, it is quite usual to encounter equations that are ill-posed and require regularization aimed at finding stable approximate solutions when the given data are noisy. In this paper, we discuss definitions and concepts for the degree of ill-posedness for linear operator equations in a Hilbert space setting. It is important to distinguish between a global version of such degree taking into account the smoothing properties of the forward operator only, and a local version combining that with the corresponding solution smoothness. We include the rarely discussed case of non-compact forward operators and explain why the usual notion of degree of ill-posedness cannot be used in this case.

Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)

MSC:

47A52	Linear operators and ill-posed problems, regularization
65J20	Numerical solutions of ill-posed problems in abstract spaces; regularization

Keywords:

degree of ill-posedness; regularisation; linear operator equation; Hilbert space; modulus of continuity; spectral distribution; source condition