Inverse Problems

Tikhonov Theory and Algorithms

Inverse problems arise in practical applications whenever one needs to deduce unknowns from observables. This monograph is a valuable contribution to the highly topical field of computational inverse problems. Both mathematical theory and numerical algorithms for model-based inverse problems are discussed in detail. The mathematical theory focuses on nonsmooth Tikhonov regularization for linear and nonlinear inverse problems. The computational methods include nonsmooth optimization algorithms, direct inversion methods and uncertainty quantification via Bayesian inference.

The book offers a comprehensive treatment of modern techniques, and seamlessly blends regularization theory with computational methods, which is essential for developing accurate and efficient inversion algorithms for many practical inverse problems.

It demonstrates many current developments in the field of computational inversion, such as value function calculus, augmented Tikhonov regularization, multi-parameter Tikhonov regularization, semismooth Newton method, direct sampling method, uncertainty quantification and approximate Bayesian inference. It is written for graduate students and researchers in mathematics, natural science and engineering.

Contents:

• Introduction
• Models in Inverse Problems
• Tikhonov Theory for Linear Problems
• Tikhonov Theory for Nonlinear Inverse Problems
• Nonsmooth Optimization
• Direct Inversion Methods
• Bayesian Inference

Readership: Advanced undergraduates, graduates and researchers in applied mathematics, computational mathematics, optimization, statistics, natural science and engineering. It will appeal to those interested in inverse problems.
Key Features:

• A large part of the materials in the book is developed by the authors, and they have not been treated in other books
• A comprehensive treatment of nonsmooth Tikhonov regularization, with a focus on value function calculus, parameter choice rules, computational algorithms, and an optimization approach to nonlinear inverse problems
• A concise introduction to fast direct methods for inverse problems, e.g., MUSIC algorithm, direct sampling method, and Gel'fand–Levitan–Marchenko transformation
• A detailed illustration of uncertainty quantification for inverse problems via Bayesian inference, including model selection, Markov chain Monte Carlo and approximate Bayesian inference