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; and serves as a convenient entry point to practical inversion. The guiding linear inversion examples are the problem of image blurring, x-ray tomography, and backward parabolic problems, including heat transfer; 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. 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.

Naíve reconstruction and inverse crimes -- Ill-posedness in inverse problems -- Truncated singular value decomposition -- Tikhonov regularization -- Total variation regularization -- Besov space regularization using wavelets -- Discretization-invariance -- Practical X-ray tomography with limited data -- Projects -- Nonlinear inverse problems -- Nonlinear inversion -- Electrical impedance tomography -- Simulation of noisy EIT data -- Complex geometrical optics solutions -- A regularized D-bar method for direct EIT -- Other direct solution methods for EIT -- Projects -- Banach spaces and Hilbert spaces -- Mappings and compact operators -- Fourier transform and Sobolev spaces -- Iterative solution of linear equations.