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Statistical and computational inverse problems. (English) Zbl 1068.65022
Applied Mathematical Sciences 160. New York, NY: Springer (ISBN 0-387-22073-9/hbk). xvi, 339 p. (2005).
The book is devoted to the development of the statistical approach to inverse problems with an emphasis on modeling and computation. Unlike many other books dedicated to this problem, it is not about the analytic results such as questions of uniqueness of the solution of inverse problems or their a priori stability. This does not mean that the authors are not recognizing the value of such results, but the main line of the book is focused on statistical and computational aspects of inversion theory. The content of the book is constructed in the following consequence. Inverse problems and interpretation of measurements is considered in the Chapter 1. Classical regularization methods and statistical inversion theory are the subjects of the Chapters 2 and 3. Nonstationary inverse problems are discussed in the Chapter 4. Classical statistical methods are the subject of the Chapter 5. Chapter 6 is devoted to the model problems. It is important to be mentioned, that in Chapters 1–6 the authors are mainly referring to books for further reading and do not discuss historical development of the topics.
In the Chapter 7 (“Case studies”), some previously known results are discussed and some new research topics are considered. It does not contain reviews of the applications. In the examples given in this chapter, some nontrivial problems are studied and new results (that either are not published) are given.
In fact, Chapters 5–7, are forming the specifics of the book. The content of this part is focused on scope of problems, for which the models for measurement errors, errorless observations and the unknown are really taken as models, which themselves may contain uncertainties.
The content is written clearly and without citations in the main text. Every chapter has a section called “Notes and comments” where the citations and further reading, as well as brief comments on more advanced topics, are provided. The book is aimed at postgraduate students in applied mathematics as well as at engineering and physics students with a firm background in mathematics. The Chapters 1–4 can be used as the material for a first course on inverse problems with a focus on computational and statistical aspects. Also, Chapters 3 and 4, which discuss statistical and non-stationary inversion methods, can be used by students already having knowledge of classical inversion methods. The inclusion of the chapter discussing the most commonly used regularization schemes provides good interpretation and analyze of those methods from the Bayesian point of view. This helps to reveal what sort of implicit assumptions these schemes are based on. The book also will be of interest for many researchers and scientists working in the area of image processing.

MSC:
65C60 Computational problems in statistics (MSC2010)
93E10 Estimation and detection in stochastic control theory
93E11 Filtering in stochastic control theory
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
62Mxx Inference from stochastic processes
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