Linear and nonlinear inverse problems with practical applications

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; 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.

pt. 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 -- pt. 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.