Linear and Nonlinear Inverse Problems with Practical Applications

Abstract

Inverse problems arise in practical applications whenever there is a need to interpret indirect measurements. This book explains how to identify ill-posed inverse problems arising in practice and how to design computational solution methods for them; explains computational approaches in a hands-on fashion, with related codes available on a website; and serves as a convenient entry point to practical inversion. The guiding linear inversion examples are the problem of image deblurring, x-ray tomography, and backward parabolic problems, including heat transfer, and electrical impedance tomography is used as the guiding nonlinear inversion example. The book s nonlinear material combines the analytic-geometric research tradition and the regularization-based school of thought in a fruitful manner, paving the way to new theorems and algorithms for nonlinear inverse problems. Furthermore, it is the only mathematical textbook with a thorough treatment of electrical impedance tomography, and these sections are suitable for beginning and experienced researchers in mathematics and engineering. Audience: Linear and Nonlinear Inverse Problems with Practical Applications is well-suited for students in mathematics, engineering, physics, or computer science who wish to learn computational inversion (inverse problems). Professors will find that the exercises and project work topics make this a suitable textbook for advanced undergraduate and graduate courses on inverse problems. Researchers developing large-scale inversion methods for linear or nonlinear inverse problems, as well as engineers working in research and development departments at high-tech companies and in electrical impedance tomography, will also find this a valuable guide. Contents Part I: Linear Inverse Problems; Chapter 1: Introduction; Chapter 2: Nave Reconstructions and Inverse Crimes; Chapter 3: Ill-Posedness in Inverse Problems; Chapter 4: Truncated Singular Value Decomposition; Chapter 5: Tikhonov Regularization; Chapter 6: Total Variation Regularization; Chapter 7: Besov Space Regularization Using Wavelets; Chapter 8: Discretization-Invariance; Chapter 9: Practical X-ray Tomography with limited data; Chapter 10: Projects; Part II: Nonlinear Inverse Problems; Chapter 11: Nonlinear Inversion; Chapter 12: Electrical Impedance Tomography; Chapter 13: Simulation of Noisy EIT Data; Chapter 14: Complex Geometrical Optics Solutions; Chapter 15: A Regularized D-bar Method for Direct EIT; Chapter 16: Other Direct Solution Methods for EIT; Chapter 17: Projects; Appendix A: Banach Spaces and Hilbert Spaces; Appendix B: Mappings and Compact Operators; Appendix C: Fourier Transforms and Sobolev Spaces; Appendix D: Iterative Solution of Linear Equations