Elements of the theory of inverse problems
``The development of the theory of inverse problems is mainly motivated by applied researches. Inverse problems arise when processing and interpreting observations in different branches of science, in particular, in geophysics. The publications in the theory of inverse problems are very numerous. Many results are presented in the monographs designed mostly for those who have had ample scientific experience. At the same time, the number of educational books introducing the readers to the theory of inverse problems is rather small.'' (From the Preface.) It is a merit of Denisov's book to present such an introduction of the theory of inverse and ill-posed problems based on lectures given by the author to the students of the Moscow State University. The author was successful in making the book accessible also to readers, who are non-specialists in this specific topic. A considerable number of exercises at the end of every chapter can help to gain a better understanding of the details. Unfortunately, there are no solutions or at least hints for the solution of these exercises at the end of the book. In Chapter 1 a series of examples of inverse problems in differential and integral form is given. Moreover, Hadamard's concept of well-posed and ill-posed problems is formulated. Well-posed and ill-posed situations and the role of associated spaces are distinguished by means of examples. Basic concepts of setting and solving inverse problems are mentioned. The Chapter 2 deals with inverse problems for ordinary differential equations in a rather comprehensive manner. Right-hand sides and coefficients of linear differential equations are to be determined. The discussion includes inverse problems for first- and second-order equations with a parameter and inverse problems for nonlinear ordinary differential equations. Chapter 3 of the book considers linear inverse problems for partial differential equations. The problems are focused on the determination of initial conditions, boundary conditions and source functions for the heat equation, the vibration equation, the Laplace equation, and the Poisson equation. Such inverse problems are reformulated as linear Fredholm or Volterra integral equations. The aim of Chapter 4 is the identification of material parameters occurring in partial differential equations. In these problems the unknown coefficient functions of the equations are to be determined using an additional information on the solution of the partial differential equation. In contrast to the inverse problems of Chapter 3 here the identification problems are nonlinear. Chapter 5 deals with the restoration of functions of one or two variables from a set of integrals of this function. A first representative of this family of inverse problems is the moment problem. Moreover, the basic problems of computerized tomography belong to that family. The book considers both classes of problems. We find assertions on the moment problem, on the inversion of the Radon transform and on the determination of a function of two variables from its integrals along circles. Finally, Chapter 6 is devoted to specific adapted solution methods for inverse problems which take into account the ill-posedness of the problems. The author presents basic knowledge on the quasi-solution method, the method of Tikhonov regularization, the iterative and the projective method of solving first kind equations and the quasi-inversion method. Furthermore, the regularization theory is applied in detail to the solution of Fredholm and Volterra integral equations of the first kind. The bibliography presented on eight pages is an interesting but not very systematic selection of books and papers on inverse and ill-posed problems providing a starting point for further studies. Unfortunately, essential books recently published in USA and Central Europe (e.g. \textit{H. W. Engl}, \textit{M. Martin} \textit{A. Neubauer}, ``Regularization of inverse problems'', Kluwer, Dordrecht (1996; Zbl 0859.65054)) are completely missing. At the end the reviewer should emphasize the authors' sentence: ``Longstanding scientific contacts with academician A. N. Tikhonov were of inestimable value in writing of this book''.
examples of inverse problems in differential and integral form
inverse problems for ordinary differential equations
inverse problems for partial differential equations
scientific article; zbMATH DE number 1414345