Operator-theoretic and regularization approaches to ill-posed problems
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 publications, partially written with coauthors, in the references. The article also reminds the reader of some valuable concepts in inverse problem theory developed by Nashed, for example, Nashed's ill-posedness definition and classification, first published in 1987, and distinguishing ill-posedness of type I and type II. On the other hand, the first author is the doyen of Geomathematics in Germany. He has founded the International Journal of Geomathematics and managed the Handbook of Geomathematics. Many interesting and challenging applications of regularization theory come from the field of geosciences and remote sensing by satellites. The comprehensive article consists of four chapters. In Chapter~1, the authors announce in a short introduction their philosophy with respect to ill-posed problems and regularization, and they list there journals, the history of textbooks, and important articles of this field. Chapter~2, entitled ``Solvability of ill-posed operator equations'', gives a short overview of linear inverse problems formulated in infinite-dimensional spaces, their discretization and regularization. The generalized inverse, a main topic of Nashed's research, plays a relevant role there. Classical regularization methods like Tikhonov regularization, truncated singular value regularization, and asymptotic regularization are mentioned. Moreover, the standard techniques for analyzing linear ill-posed problems with compact forward operators by singular values and Fourier expansions are introduced. Chapter~3, entitled ``Operator methodologies of resolution'', discusses the philosophy and methods for obtaining stable approximate solutions to ill-posed problems. The authors distinguish between ``solution'' of an operator equation and ``resolution'' for an inverse problem. With respect to the instability effect coming from ill-posedness and chances for overcoming it or at least its reduction by exploiting additional a~priori information, the computational standpoint is relevant for the authors. In this chapter and in the subsequent Chapter~4, entitled ``Reconstruction methods and regularizing filters'', they focus on three aspects of the problem area: (a) strategy of resolution and reconstruction, (b) regularization-approximation schema, (c) regularization algorithms. In Chapter~3, the concept of the authors concerning regularization with occurring error bounds are outlined. Moreover, the use of compactness and a priori bounds is discussed. Tikhonov's regularization method, the role of source conditions and convergence rates are also briefly under consideration. The chapter is concluded by the characterization of regularizers. This, however, requires a deep analysis of linear operators in Banach and Hilbert spaces, the role of their null-spaces and ranges, and of associated generalized inverses. Reconstruction methods based on regularizing filter factors form the content of the final Chapter~4, encompassing more than 60 pages. Assuming that the forward operator of the inverse problem is a bounded and compact linear operator mapping between infinite-dimensional Hilbert spaces with non-closed range, preferably linear regularization procedures are analyzed in eight sections of this chapter. In an introductory prologue of 16 pages, the basics of the theory of linear regularization methods are outlined, meeting optimality concepts, worst case errors, regularizing filters, the choice of regularization parameters, source conditions, and convergence rates. The Tikhonov-Phillips regularization is discussed in detail as an important example for exercising all these theory components. In Section~4.1, generalized inverses in reproducing kernel Hilbert spaces are presented and used in order to construct appropriate regularization techniques. For approaching the computational aspects, discretization is necessary and Section~4.2 follows this requirement by studying projection methods. On the other hand, Section~4.3 turns to multiscale methods as regularization schemes. Again with respect to computational aspects and as alternative to variational regularization, iterative regularization methods are in the focus of Section~4.4. It is unavoidable in regularization theory to take into account also stochastic aspects with respect to noisy data or a randomization of the solution in a Bayesian setting. Section~4.5 briefly meets such concepts. Mollifier methods are considered in Section~4.6, and the Backus-Gilbert method is discussed in Section~4.7. The commendable Section~4.8 on numerical dilemmas and methodologies completes Chapter~4.
scientific article; zbMATH DE number 6915836
scientific article