Operator-theoretic and regularization approaches to ill-posed problems
type: Item
instance of: scholarly article
title: Operator-theoretic and regularization approaches to ill-posed problems
zbMATH Open document ID: 1464.47006
author: Willi Freeden
author: M. Zuhair Nashed
published in: GEM - International Journal on Geomathematics
publication date: 2018-08-10T00:00:00Z
review text: With this survey paper, the authors, both experts in the field of inverse and ill-posed problems, present an excellent overview of the history and new developments in the context of operator-theoretic and regularization approaches to ill-posed problems. Although the paper is published as a journal article, it can be considered as a textbook on regularization theory and practice with focus on its functional analytic background. It is hoped that numerous mathematicians, young scientists, and practitioners who are dealing with inverse problems and with the phenomenon of ill-posedness take note of this outstanding work. The readers will also find the comprehensive bibliography with more than 250 references very useful. The paper is highly recommended to all who want to understand the philosophy of resolution with respect to inverse problems which are expressed by linear and nonlinear operator equations formulated in abstract Banach, Hilbert, or metric spaces. Moreover, the interconnections between the formulation of the problems in infinite-dimensional spaces with proper ill-posedness and its discrete analogues with ill-conditioning phenomena as immediate consequences of the underlying ill-posedness are outlined. For any ill-posed applied inverse problem in natural sciences, engineering, or finance, the practitioner has to decide whether first to regularize the operator equation and then discretize it or, alternatively, first to discretize the problem and then regularize the finite-dimensional system is the better and more appropriate approach. This, however, requires a deep understanding of the operator-theoretic basics, and the paper under review can help to achieve this knowledge. The second author is the doyen of operator theory with respect to ill-posed problems. From the reviewer's point of view, some chapters of the article can be seen in some sense as a mathematical legacy of Nashed's achievements over almost 50 years. One can find more than 25 of his publicatio…
reviewed by: Bernd Hofmann
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 86A22
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 6915836
zbMATH Keywords: ill-posed problems
zbMATH Keywords: inverse problems
zbMATH Keywords: regularization
zbMATH Keywords: generalized inverses
MaRDI profile type: Publication
full work available at URL: s13137-017-0100-0
OpenAlex ID: W2774066893
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Operator-theoretic and regularization approaches to ill-posed problems (DOI)
DOI: 10.1007/S13137-017-0100-0