Linear and Nonlinear Inverse Problems with Practical Applications

A guide to inverse problems and the computational techniques used to solve them, suitable for graduate students and researchers.

Jennifer L. Mueller (Author), Samuli Siltanen (Author)

9781611972337

Paperback / softback, published 30 November 2012

364 pages
22.9 x 15.2 x 2 cm, 0.635 kg

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.

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.

Subject Areas: Miscellaneous items [WZ]