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2021 | Buch

Introduction and Basic Concepts Kapitel lesen Erstes Kapitel lesen

An Introduction to the Mathematical Theory of Inverse Problems

verfasst von: Andreas Kirsch

Verlag: Springer International Publishing

Buchreihe : Applied Mathematical Sciences

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

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SUCHEN

Über dieses Buch

This graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering.
This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.

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.
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.
Andreas Kirsch
Chapter 4. Nonlinear Inverse Problems
Abstract
In the previous chapters, we considered linear problems which we wrote as \(Kx=y\), where K was a linear and (often) compact operator between Hilbert spaces. Needless to say that most problems in applications are nonlinear. For example, even in the case of a linear differential equation of the form \(-u^{\prime \prime }+cu=f\) for the function u the dependence of u on the parameter function c is nonlinear; that is, the mapping \(c\mapsto u\) is nonlinear. In Chapters 5, 6, and 7 we will study particular nonlinear problems to determine parameters of an ordinary or partial differential equation from the knowledge of the solution.
Andreas Kirsch
Chapter 5. 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 [148]. Other applications appear in parameter identification problems for parabolic or hyperbolic differential equations—as we study in Section 5.6 for a model problem—(see also [167, 187, 255]) or in grating theory ([156]).
Andreas Kirsch
Chapter 6. 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.
Andreas Kirsch
Chapter 7. An Inverse Scattering Problem

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), \(\rho (x,t)\), 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. Then the movement of the particle is described by the following equations.

Andreas Kirsch
Backmatter
Metadaten
Titel
An Introduction to the Mathematical Theory of Inverse Problems
verfasst von
Andreas Kirsch
Copyright-Jahr
2021
Electronic ISBN
978-3-030-63343-1
Print ISBN
978-3-030-63342-4
DOI
https://doi.org/10.1007/978-3-030-63343-1

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