**Description:**

This
book is an introduction to both computational inverse problems and uncertainty
quantification (UQ) for inverse problems. The book also presents more advanced
material on Bayesian methods and UQ, including Markov chain Monte Carlo
sampling methods for UQ in inverse problems. Each chapter contains MATLAB® code
that implements the algorithms and generates the figures, as well as a large
number of exercises accessible to both graduate students and researchers.

*Computational Uncertainty Quantification for Inverse
Problems* is intended for graduate students, researchers, and
applied scientists. It is appropriate for courses on computational inverse
problems, Bayesian methods for inverse problems, and UQ methods for inverse
problems.

Contents:

**Preface**

**Chapter 1: Characteristics of Inverse Problems** •
Preliminaries • The least squares estimator • The statistical properties of x_{LS}
and ill-posedness • An illustrative example • Exercises

**Chapter 2: Regularization by Spectral Filtering** •
Spectral filtering methods • Regularization parameter selection methods •
Periodic and data-driven boundary conditions • Exercises

**Chapter 3: Two-Dimensional Test Cases** •
Two-dimensional image deblurring • Computed tomography • The preconditioned
conjugate gradient iteration • Exercises

**Chapter 4: Bayes’ Law, Markov Random Field Priors, and MAP
Estimation** • Bayes’ law and
regularization • Choosing *p*(x|*d*): Gaussian Markov random fields •
Choosing *p*(x|*d*): Laplace Markov random fields • The
infinite-dimensional limit • Exercises

**Chapter 5: Markov Chain Monte Carlo Methods for Linear
Inverse Problems** • Sampling from high-dimensional Gaussian
random vectors • Hierarchical modeling of ? and d and sampling from *p*(x,
*?*, *d*|b) • Alternative MCMC methods for sampling from *p*(x, *?*,
*d*|b) • Exercises

**Chapter 6: Markov Chain Monte Carlo Methods for Nonlinear
Inverse Problems** • A general setup for nonlinear inverse
problems • Levenburg–Marquardt nonlinear least squares optimization •
Randomize-then-optimize as a proposal for Metropolis–Hastings • Nonlinear test
cases • Hierarchical modeling of *?* and *d* and sampling from *p*(x,
*?*, *d*|b) • Exercises

**Bibliography**

**Index**

About the Author:

**Johnathan M. Bardsley** is a professor in the
Department of Mathematical Sciences at the University of Montana, where he has
been teaching since 2003. He has held long-term visiting professorships at the
University of Helsinki, Finland; University of Otago, New Zealand; Technical
University of Denmark; and Monash University, Australia, supported by the
Gordon Preston Sabbatical Fellowship. Professor Bardsley was a postdoctoral
fellow at the NSF-funded Statistical and Applied Mathematical Sciences Institute
during its inaugural year in 2002–03. In 2017, he received the Chancellor’s
Medallion Award from Montana Tech for excellence in his educational and
professional career and for significant contributions to his academic
discipline. Professor Bardsley’s research interests focus on inverse problems,
uncertainty quantification, computational mathematics, and computational
statistics, and he has published many refereed journal articles in these areas.

Target Audience:

This
book is intended for graduate students, researchers, and applied scientists. It
is appropriate for courses on computational inverse problems, Bayesian methods
for inverse problems, and UQ methods for inverse problems.