An Introduction to the Mathematical Theory of Inverse Problems

1. Front Matter

   Pages i-xvii

   PDF

2. Introduction and Basic Concepts

   • Andreas Kirsch

   Pages 1-21

3. Regularization Theory for Equations of the First Kind

   • Andreas Kirsch

   Pages 23-61

4. Regularization by Discretization

   • Andreas Kirsch

   Pages 63-118

5. Nonlinear Inverse Problems

   • Andreas Kirsch

   Pages 119-168

6. Inverse Eigenvalue Problems

   • Andreas Kirsch

   Pages 169-214

7. An Inverse Problem in Electrical Impedance Tomography

   • Andreas Kirsch

   Pages 215-238

8. An Inverse Scattering Problem

   • Andreas Kirsch

   Pages 239-313

9. Back Matter

   Pages 315-400

   PDF

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.  

• Inverse Problems
• ill-posed problems
• regularization techniques
• scattering theory
• inverse spectral problems
• impedance tomography
• Tikhonov regularization
• Landweber's method
• conjugate gradient method

“This monograph is a thorough and insightful introduction to the mathematics of inverse problems and a solid improvement of the previous editions, which were used to educate many researchers in the field over the last two and a half decades. As such, the volume is already a classic and can be recommended without reservations to any reader interested in both the foundations and specific examples of inverse problems relevant to modern engineering and sciences.” (Alexander Mamonov, SIAM Review, Vol. 65 (2), 2023)

• Department of Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

  Andreas Kirsch

Andreas Kirsch is a Professor in the Department of Mathematics at Karlsruhe Institute of Technology.

Policies and ethics