This is yet another book in the successful series “Fundamentals of Algorithms” from the Society for Industrial and Applied Mathematics (SIAM). It introduces the tools of modern applied mathematics in a style accessible to nonmathematicians. The key in this series is to focus on a small but representative class of basic applications, in order to motivate mathematical innovations and inspire the interested reader to consult the numerous references cited to learn about the scope of the described computational tools.

The book follows this style by concentrating on a single class of boundary value problems, those that can be modeled by one-dimensional Fredholm integral equations of the first kind:

∫₀¹ K(s,t) f(t) dt = g(s).

These equations appear widely, in many branches of physics. Their mathematical significance lies in their ability to describe boundary value problems in a global sense; that is, the statement of the problem relies on the evaluation of the integral in (1) over the entire domain (0, 1). The forward problem assumes knowledge of the kernel K, and asks for the determination of f. The inverse problem requires computing K from partial and possibly noisy knowledge of f and g.

Inverse problems are notoriously hard, analytically and computationally, because they are ill posed by their very nature--small perturbations of the measured data often result in large perturbations in the determination of the desired parameters. The author dedicates the first chapter to getting this point across. Then, he goes further to illustrate how important this observation is in the process of actually computing a solution. The subsequent chapters dive into methods that have proven very successful in the past few decades, by first motivating these established algorithms on toy problems to build intuition and confidence, before embarking on more realistic applications. The main topics covered are singular value decomposition, regularization methods, and iterative methods. Among the applications considered in detail are the barcode reading problem and image deblurring. Reminiscent of the remarkably successful book by Nick Trefethen [1], this book is supplemented by a valuable collection of MATLAB codes that will add quite a bit of pleasure to the reader’s experience.

This book should serve as an excellent resource for advanced undergraduates and beginning graduate students in all fields of science and engineering. Its style should also serve as a guide for students who are contemplating writing a thesis. Although not exhaustive or complete, its content could play a key role in introducing to nonmathematicians one of the most active disciplines in applied mathematics.