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Inverse problems in the mathematical sciences. (English) Zbl 0779.45001

Vieweg Mathematics for Scientists and Engineers. Braunschweig: Vieweg. v, 152 p. (1993).
Given a model (“black box”) \(K\), the direct problem associated with this model consists in determining all possible unknown effects (“outputs”) \(y=Kx\) from all possible known causes (“inputs”) \(x\). From the physical viewpoint, however, so-called inverse problems are more natural and interesting. Such inverse problems may be stated in two different ways: one may try to recover \(x\) from \(K\) and \(y\) (“causation problem”), or \(K\) from \(x\) and \(y\) (“model identification problem”).
There is a vast literature on very special inverse problems, as well as some monographs on selected topics, for example, the nice treatment on the inverse spectral problem for Hill’s equation with Dirichlet boundary conditions by J. Pöschel and E. Trubowitz [Academic Press, New York (1987; Zbl 0623.34001)]. On the other hand, there is a deplorable lack of introductory self-contained books on both the theory and applications of inverse and ill-posed problems.
The present book aims to fill this gap, and so it does with great success. It consists of an introduction, four chapters, and an annotated bibliography. The first two chapters are concerned with the a.m. causation and identification problem, respectively, in the mathematical modelling of various physical phenomena, with an emphasis on Fredholm integral equations of the first kind. In the following chapter the author discusses the functional-analytic background for such problems, covering function spaces, operator theory, spectral properties, and generalized inverses. Some methods which are particularly important for “solving” inverse and ill-posed problems (e.g., the Tikhonov-Phillips regularization method and the Backus-Gilbert method) are discussed in the final chapter. One of the most useful parts of the book is the bibliography, where the author gives not just a list of references, but also a brief summary of the contents and style of each of these references.
This is a masterful work on the theory, some methods, and various applications of both inverse and ill-posed problems. The presentation is very clear, the style excellent, and the mathematics sufficiently simple. The author also provides some historical comments (“wie es eigentlich gewesen”), as well as many elementary exercises. This really nice book deserves a large readership among researchers, teachers, students, and engineers interested in this field of still growing importance.

MSC:

45-02 Research exposition (monographs, survey articles) pertaining to integral equations
45B05 Fredholm integral equations
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65R30 Numerical methods for ill-posed problems for integral equations
45P05 Integral operators
35R25 Ill-posed problems for PDEs
34A55 Inverse problems involving ordinary differential equations
35R30 Inverse problems for PDEs

Citations:

Zbl 0623.34001
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