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2011 | Buch

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An Introduction to the Mathematical Theory of Inverse Problems

verfasst von: Andreas Kirsch

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book introduces the reader to the area of inverse problems. The study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography.
The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples.
The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed.

In this new edition, the Factorization Method is included as one of the prominent members in this monograph. Since the Factorization Method is particularly simple for the problem of EIT and this field has attracted a lot of attention during the past decade a chapter on EIT has been added in this monograph as Chapter 5 while the chapter on inverse scattering theory is now Chapter 6.The main changes of this second edition compared to the first edition concern only Chapters 5 and 6 and the Appendix A. Chapter 5 introduces the reader to the inverse problem of electrical impedance tomography.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Basic Concepts

Abstract

In this section, we present some examples of pairs of problems that are inverse to each other. We start with some simple examples that are normally not even recognized as inverse problems. Most of them are taken from the survey article [136] and the monograph [99].

Andreas Kirsch

Chapter 2. Regularization Theory for Equations of the First Kind

Abstract

We saw in the previous chapter that many inverse problems can be formulated as operator equations of the form

$$Kx = y,$$

where K is a linear compact operator between Hilbert spaces X and Y over the field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We also saw that a successful reconstruction strategy requires additional a priori information about the solution.

Andreas Kirsch

Chapter 3. Regularization by Discretization

Abstract

In this chapter, we study a different approach to regularizing operator equations of the form Kx = y, where x and y are elements of certain function spaces. This approach is motivated by the fact that for the numerical treatment of such equations one has to discretize the continuous problem and reduce it to a finite system of (linear or nonlinear) equations. We see in this chapter that the discretization schemes themselves are regularization strategies in the sense of Chap. 2.

Andreas Kirsch

Chapter 4. Inverse Eigenvalue Problems

Abstract

Inverse eigenvalue problems are not only interesting in their own right but also have important practical applications. We recall the fundamental paper by Kac [132]. Other applications appear in parameter identification problems for parabolic or hyperbolic differential equations (see [149, 170, 234]) or in grating theory ([140]).

Andreas Kirsch

Chapter 5. An Inverse Problem in Electrical Impedance Tomography

Abstract

Electrical impedance tomography (EIT) is a medical imaging technique in which an image of the conductivity (or permittivity) of part of the body is determined from electrical surface measurements. Typically, conducting electrodes are attached to the skin of the subject and small alternating currents are applied to some or all of the electrodes. The resulting electrical potentials are measured, and the process may be repeated for numerous different configurations of applied currents.

Andreas Kirsch

Chapter 6. An Inverse Scattering Problem

Abstract

We consider acoustic waves that travel in a medium such as a fluid. Let v(x, t) be the velocity vector of a particle at $x \in {\mathbb{R}}^{3}$ and time t. Let p(x, t), ρ(x, t), and S(x, t) denote the pressure, density, and specific entropy, respectively, of the fluid. We assume that no exterior forces act on the fluid.

Andreas Kirsch

Backmatter

Titel: An Introduction to the Mathematical Theory of Inverse Problems
verfasst von: Andreas Kirsch
Verlag: Springer New York
Electronic ISBN: 978-1-4419-8474-6
Print ISBN: 978-1-4419-8473-9
DOI: https://doi.org/10.1007/978-1-4419-8474-6

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