Inverse problems are frequently encountered in real-world applications. Ill-posed (ill-conditioned) problems, or equivalently, singular problems are often associated with them. As the name suggests, this kind of problem is usually inexact, and cannot be dealt with directly; only partial or indirect information is available, and reconstruction can only be performed from noisy data.

A broad range of material is covered in this book, which provides a solid introduction to the field of inverse problems, and techniques for their solution: regularization methods and optimal estimation. The main topics covered include numerical optimization techniques, error analysis, and rates of convergence. Also covered are regularization parameter selection methods, the total variation regularization method, and constrained optimization methods.

Chapter 1 motivates the study of inverse problems, and introduces some regularization methods. It also outlines two important topics: error analysis, which tells us how accurate a solution we can get for a specific method; and rate of convergence, which tells us how fast we can get to the converged solution, if some specific solution procedure is to be applied.

Chapters 2 and 3 present mathematical tools that will be used throughout the book to analyze and solve ill-posed problems. Chapter 2 is presented from an analytical perspective, and chapter 3 is more numerical.

Chapter 4 presents an introduction to statistical estimation theory. As is often the case, practical real-world applications involve measurement, which is never perfect; noisy data is always expected. Statistical estimation theory is particularly useful when the statistical model (for example, Gaussian or Poisson) is built. Best fit-to-data, in the sense of least squares, is to be achieved.

Chapter 5 presents the image deblurring problem and solution techniques. This is a typical two-dimensional deconvolution/reconstruction problem. Chapter 6 addresses parameter identification problems. Special structures arise when discretization, which is often needed in real applications, is applied to distributed parameter systems. The approximation of gradient and Hessian computations is also discussed, from the perspective of implementation efficiency.

Chapter 7 discusses the issue of how to select the regularization parameter properly. One often encounters the problem of too much or too little regularization when performing reconstruction on given noisy data. This naturally leads one to seek a regularized solution that minimizes some indicator of solution fidelity, namely some sensible measure of estimation error. This chapter emphasizes methods that make use of statistical information about the noise in the data. Chapter 8 specifically discusses the total variation regularization method.

Chapter 9 presents constrained optimization methods. Practical inverse problems may involve some quantities that have physical meanings that cannot be negative, for example, mass, volume, probability function, or image density. This motivates the formulation of constrained optimization problems. From an optimization perspective, one wants to stay inside the feasible set of solutions. Another important motivation for constraint optimization is the fact that imposing a priori constraints can sometimes improve the solution dramatically.

A lot of material on numerical algorithms is covered in this book, which, from an implementation point of view, will be very useful for practicing researchers and engineers. I found the book to be very readable and well written.