Computational Uncertainty Quantification for Inverse Problems

Computational Uncertainty Quantification for Inverse Problems is a great reference book on inverse problems that could  also be adopted for a graduate-level textbook targeting Ph.D. candidates.  It covers the introduction to computational inverse problems in the first part, and includes various sampling methods for uncertainty quantification in inverse problems in the second part of the book. 

 

This book utilizes many components from computational mathematics and computational statistics to implement uncertainty quantification in inverse problems, such as least squares, regularized optimization, preconditioned conjugate gradients, Fourier transforms, Markov random fields, maximum a posteriori, Markov chain Monte Carlo (MCMC) sampling methods, Metropolis-Hastings algorithm, etc. The book devotes a relatively large amount of space to the derivation of formulas. However, due to the length of the book, many details are left as exercises or for the readers to figure out. Computer science and engineering students would appreciate seeing more algorithms, and mathematics students might like to see more of the basics of the included mathematics and statistics components.

 

One aspect expected by graduate students is the application of the computational methods. The author has provided application examples with the results of comparing different methods. A collection of MATLAB codes that implement the algorithms (which are also referenced in the book) are available to download on the SIAM website. 

 

The book is quite comprehensive. It covers topics in inverse problems and focuses on uncertainty quantification in inverse problems without providing too many details. The chapters are well organized. The instability of the traditional least squares estimator for inverse problems is first discussed in Chapter 1, and regularization methods are introduced in Chapter 2 to improve least-squares solutions for one-dimensional test cases. Then the methods are extended to two-dimensional test cases and conjugate gradient iteration is discussed for computed tomography in Chapter 3. In Chapters 4-6, an uncertainty quantification component is included for inverse problems. Specifically, prior modeling using Markov random fields for implementing a Bayesian statistical framework is introduced for the inverse problem, which is followed by either maximizing the posterior density function for the maximum a posteriori (MAP) estimator or using various sampling methods (such as hierarchical Gibbs sampling, Metropolis-Hastings algorithms, randomize-then-optimize proposal density for the Metropolis-Hastings MCMC method) to sample posterior density function with or without hyper-priors. Some of the approaches that the author introduces for uncertainty quantification are from his recent publications.  

 

Overall, the author succeeds in introducing the computational methods for inverse problems and their uncertainty quantification aspects. In addition, the author has provided a collection of MATLAB codes to implement the algorithms for application examples. 

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Dr. Yanyan He is an assistant professor of Mathematics and Computer Science at the University of North Texas.