A Taste of Inverse Problems: Basic Theory and Examples

Inverse problems need to be solved in order to properly interpret indirect measurements. Often, inverse problems are ill-posed and sensitive to data errors. Therefore one has to incorporate some sort of regularization to reconstruct significant information from the given data.

This book presents the main achievements that have emerged in regularization theory over the past 50 years, focusing on linear ill-posed problems and the development of methods that can be applied to them. Some of this material has previously appeared only in journal articles.

A Taste of Inverse Problems: Basic Theory and Examples rigorously discusses state-of-the-art inverse problems theory, focusing on numerically relevant aspects and omitting subordinate generalizations; presents diverse real-world applications, important test cases, and possible pitfalls; and treats these applications with the same rigor and depth as the theory.

Audience: The book is intended for graduate students and researchers, and it is suitable for engineers who are faced with solving specific inverse problems. The prerequisites for this book are undergraduate-level mathematics and a basic knowledge of elementary Hilbert space theory.

Contents: Preface; Chapter 1: Amuse-Gueule: Numerical Differentiation; Part I: Ill-Posed Problems; Chapter 2: Compact Operator Equations; Chapter 3: The Hausdorff-Tikhonov Lemma; Chapter 4: Computerized X-Ray Tomography; Chapter 5: The Cauchy Problem for the Laplace Equation; Chapter 6: The Factorization Method in EIT; Part II: Tikhonov Regularization; Chapter 7: Tikhonov s Method; Chapter 8: The Discrepancy Principle; Chapter 9: Smoothing Splines; Chapter 10: Deconvolution and Imaging; Chapter 11: Evaluation of Unbounded Operators; Part III: Iterative Regularization; Chapter 12: Landweber Iteration; Chapter 13: The Discrepancy Principle; Chapter 14: Iterative Solution of the Cauchy Problem; Chapter 15: The Conjugate Gradient Method; Chapter 16: The Gerchberg-Papoulis Algorithm; Chapter 17: Inverse Source Problems; Appendices: A: The Singular Value Decomposition; B: Sobolev Spaces; C: The Poisson Equation; D: Sobolev Spaces (cont.); E: The Fourier Transform; F: The Conjugate Gradient Iteration; Bibliography; Index.

Martin Hanke is a Professor of Mathematics at the Johannes Gutenberg-University in Mainz, Germany. He works in numerical analysis and his research focuses on inverse and ill-posed problems, the development of general regularization methods, and the analysis of sophisticated algorithms for specific inverse problems.