Methods and Applications of Analysis

Volume 17 (2010)

Number 4

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Pages: 445 – 462

DOI: http://dx.doi.org/10.4310/MAA.2010.v17.n4.a8

Authors

Bernd Hofmann

Stefan Kindermann

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.

Keywords

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

2010 Mathematics Subject Classification

47A52, 65J20

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