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Full Description
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 gives a hands-on guide to designing computational solution methods for them, with related codes on an accompanying website. The guiding linear inversion examples are the problem of image deblurring, x-ray tomography, and backward parabolic problems, including heat transfer. A thorough treatment of electrical impedance tomography is used as the guiding nonlinear inversion example which combines the analytic-geometric research tradition and the regularization-based school of thought in a fruitful manner. This book is complete with exercises and project topics, making it ideal as a classroom textbook or self-study guide for graduate and advanced undergraduate students in mathematics, engineering or physics who wish to learn about computational inversion. It also acts as a useful guide for researchers who develop inversion techniques in high-tech industry.
Contents
Part I: Linear Inverse Problems
Chapter 1: Introduction
Chapter 2: Naïve 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
