Elements of the theory of inverse problems
type: Item
instance of: scholarly article
title: Elements of the theory of inverse problems
zbMATH Open document ID: 0940.35003
author: A. M. Denisov
published in: Inverse and Ill-Posed Problems Series
publication date: 2000-03-14T00:00:00Z
review text: ``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, bou…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 65L09
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 65R32
Mathematics Subject Classification ID: 65J22
zbMATH DE Number: 1414345
zbMATH Keywords: ill-posedness parameter identification
zbMATH Keywords: examples of inverse problems in differential and integral form
zbMATH Keywords: inverse problems for ordinary differential equations
zbMATH Keywords: inverse problems for partial differential equations
zbMATH Keywords: Volterra integral equations
zbMATH Keywords: unknown coefficient functions
zbMATH Keywords: moment problem
zbMATH Keywords: Radon transform
zbMATH Keywords: quasi-solution
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: quasi-inversion
reviewed by: Bernd Hofmann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1414345