Traditionally, mathematical methods based on ordinary and partial differential equations have been used in science and engineering to predict the behavior of a system based on its physical parameters. In practice, scientists and engineers are often faced with the problem of determining the physical parameters of a system from indirect measurements of the behavior of the system. Engineers and scientists working in various disciplines have worked on these problems in isolation and have often developed similar methods with different terminology and notation.
Mathematicians have applied the tools of functional analysis to inverse problems and developed very powerful theoretical approaches, but the mathematical theory is not very accessible to engineers and scientists who have not studied analysis at the graduate level. Things become even more complicated when inverse problems are considered from a statistical point of view. There is an obvious need to synthesize the various approaches and to provide textbooks that explain the subject from different points of view while being accessible to students who may have a limited mathematical or statistical background.
The authors of this book are respectively a mathematician and an engineer. Their book attempts to introduce the subject of inverse problems in a way that emphasizes practical methods for the solution of applied problems while also providing a general understanding of the subject. In order to make the book accessible to scientists and engineers, the authors have assumed that readers are familiar only with mathematics to the level of calculus, linear algebra, and differential equations. This makes it among the most accessible textbooks on the subject.
The book begins with a set of three introductory chapters that provide an introduction to mathematical modeling and inverse problems. After this introduction the authors give a concise development of the spectral theory of discrete linear inverse problems and Tikhonov regularization. This chapter also includes several iterative methods for the regularization of discrete linear inverse problems including the Landweber iteration and the conjugate gradient method.
The second half of the book consists of largely self contained chapters on particular inverse problems, including applications to image restoration and heat transfer. These chapters build on and extend the discussion of discrete linear inverse problems. For example, in the chapter on image restoration, the authors begin with Tikhonov regularization but then go on to consider maximum entropy regularization. The resulting optimization problems are nonlinear least squares problems, so the authors introduce Newton's method and the Levenberg-Marquardt method for solving nonlinear least squares problems. In a later chapter on heat conduction, the authors go beyond the discretized formulation of the problem and consider optimization over a space of functions. They then introduce the adjoint equation and use it to obtain a gradient for use with the conjugate gradient method.
This book should be of interest to instructors who want to teach an introductory course in inverse problems with limited mathematics prerequisites to an audience of scientists and engineers at the advanced undergraduate or graduate level. As a textbook, the strengths of this book are the extensive examples and exercises, as well as an appendix covering required mathematical background. One weakness is that the authors have not made example code available. The book would also be accessible to graduate students in engineering and science for self study. The individual application oriented chapters in the second half of the book are independent enough that readers with some background in inverse problems can dip into them without having to read the entire book.
Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.