Linear inverse problems and Tikhonov regularization (Q2828460)

This text provides a clear and illustrated presentation of linear inverse problems, with a focus on Tikhonov regularization for solving ill-posed problems in Hilbert spaces. It covers important aspects like the instability of ill-posed problems, parameter choice strategies, and the singular value expansion (SVE) of linear compact operators.NEWLINENEWLINEThe text consists of five chapters. The first chapter provides a concise introduction to inverse problems. Initially, the instability in solving first-kind linear integral equations and the numerical stability of second-kind linear integral equations are examined. Afterwards, an elastic string is considered which serves as an example that a direct problem is stable with respect to perturbations, whereas the corresponding inverse problem is not stable.NEWLINENEWLINEChapter 2 introduces the concept of well-posedness of equations \( Tx = y \), where \( T: X \to Y \) is a bounded linear operator between Hilbert spaces \( X \) and \( Y \). Conditions for uniqueness and existence of solutions are given, and minimum norm solutions and least squares solutions are considered. In addition, the generalized inverse is introduced, and two-dimensional X-ray transmission tomography is considered as an example for an inverse problem. In the final section of this chapter, discretized inverse problems in form of matrix equations \( A\mathbf{x} = \mathbf{y} \) are examined.NEWLINENEWLINEChapter 3 deals with classical Tikhonov regularization of linear ill-posed problems in Hilbert spaces. It starts with a consideration of the existence of the Tikhonov solution, followed by convergence results in case of exact data. Subsequently, the regularizing properties are examined, i.e., both results on convergence and rates of convergence are presented. Other topics in this chapter are a~posteriori parameter choice strategies like the discrepancy principle and the L-curve criterion. Arbitrarily slow convergence, converse results and saturation are also considered.NEWLINENEWLINEIn Chapter 4, compact operators are introduced as a prototype for a linear ill-posed problem in Hilbert spaces. The singular value expansion (SVE) is considered as a tool to analyze the results on Tikhonov regularization. In addition, it is shown that the SVE can be used to represent the generalized inverse of a compact operator, and fractional powers of operators and another converse result are considered. Further topics in this chapter are general regularization methods, and the method of asymptotic regularization is considered as an example.NEWLINENEWLINEIn Chapter 5, some basic theory of generalized Tikhonov regularization \( \| Tx-y \|_Y^2 + \lambda \| Lx \|_Z^2 \to \min \) with an unbounded linear operator \( L : X \supset D \to Z \) is presented, where \( Z \) denotes a Hilbert space. Finally, two appendices provide the basic tools from functional analysis required for Hilbert and Sobolev spaces.NEWLINENEWLINEMany graphical illustrations, results of numerical experiments and historical notes are included in this well-organized textbook that can be used for self-study. The technical background required to read the book is relatively low, and discussions are always kept simple, which makes the book easily intelligible to a wide range of readers.