Methods and Applications of Analysis
Volume 17 (2010)
On the Degree of Ill-posedness for Linear Problems with Noncompact Operators
Pages: 445 – 462
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.
Degree of ill-posedness; regularization; linear operator equation; Hilbert space; modulus of continuity; spectral distribution; source condition
2010 Mathematics Subject Classification