Inverse Problems. Basics, theory and applied examples (Q2346865)

The textbook consists of five chapters. Chapter 1 motivates the importance of inverse problems by presenting some applications, e.g., from signal analysis, gravimetry, and computerized tomography. In addition, the basics of some mathematical tools are introduced, e.g., Banach and Hilbert spaces, operator theory, the Fourier transform, and function spaces like \( L^2 \) and Sobolev spaces on finite intervals. The definition of ill-posedness is also considered in this introductory chapter. Chapter 2 introduces the reader to finite-dimensional linear least squares problems. Existence, uniqueness, and sensitivity with respect to perturbations are investigated in this chapter. The next chapter (Chapter 3) deals with the discretization of inverse problems. After a preparatory section on spline functions, the author introduces the basics of projection methods (collocation method, Galerkin method, least squares method) for solving linear ill-posed problems. Another topic in this chapter is the discrete Fourier transform for solving convolution equations and for inverting the Radon transform. Chapter 4 addresses the regularization of linear ill-posed problems. Some basic concepts are introduced first, followed by a section on linear least squares problems with quadratic constraints and their connection with Tikhonov regularization. The discrepancy principle for Tikhonov regularization is also considered in some detail, followed by a presentation of the Landweber iteration and an implicit iteration scheme. All methods in this chapter are considered in a finite-dimensional setting which is obtained by applying some projection scheme to the original linear ill-posed problem. The final chapter of the text (Chapter 5) provides an introduction to the regularization of nonlinear ill-posed problems. It starts with a parameter estimation problem for an ordinary differential equation which results from an application in elastography, and discretization by finite elements finally leads to a finite-dimensional nonlinear least squares problem. The rest of this section is devoted to the regularization of general nonlinear equations and least squares problems. An a priori as well as an a posteriori parameter choice strategy for Tikhonov regularization is considered, followed by a short section on iteration schemes like the Gauss-Newton method. This is a clearly written and concise textbook which considers selected topics from the theory and application of inverse problems. The presentation is done on an introductory level, with the necessary prerequisites from linear algebra and functional analysis being introduced in the text. This makes the book suitable for students from mathematics and natural sciences as well. There is an emphasis on the presentation of examples from applied research, and several numerical illustrations are included.