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.
• A convenient entry point to practical inversion • Shows how to identify ill-posed inverse problems and design computational solution methods for them • Explains computational approaches in a hands-on fashion, with related codes available on a website
Contents
Part I. Linear Inverse Problems: 1. Introduction; 2. Naïve reconstructions and inverse crimes; 3. Ill-posedness in inverse problems; 4. Truncated singular value decomposition; 5. Tikhonov regularization; 6. Total variation regularization; 7. Besov space regularization using wavelets; 8. Discretization-invariance; 9. Practical X-ray tomography with limited data; 10. Projects; Part II. Nonlinear Inverse Problems: 11. Nonlinear inversion; 12. Electrical impedance tomography; 13. Simulation of noisy EIT data; 14. Complex geometrical optics solutions; 15. A regularized D-bar method for direct EIT; 16. Other direct solution methods for EIT; 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.
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