Methods and Applications of Analysis

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

Bernd Hofmann and Stefan Kindermann

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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.

Article information

Source
Methods Appl. Anal., Volume 17, Number 4 (2010), 445-462.

Dates
First available in Project Euclid: 24 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.maa/1306249562

Mathematical Reviews number (MathSciNet)
MR2800562

Zentralblatt MATH identifier
1228.47017

Subjects
Primary: 47A52: Ill-posed problems, regularization [See also 35R25, 47J06, 65F22, 65J20, 65L08, 65M30, 65R30] 65J20: Improperly posed problems; regularization

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

Citation

Hofmann, Bernd; Kindermann, Stefan. On the Degree of Ill-posedness for Linear Problems with Noncompact Operators. Methods Appl. Anal. 17 (2010), no. 4, 445--462. https://projecteuclid.org/euclid.maa/1306249562


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