The evaluation of integrals and solution of partial differential equation boundary value problems are commonplace in many areas of science and engineering. Associated with these problems are inverse problems in which we use observations of a system to recover information about source terms or boundary conditions or the coefficients of a PDE. There are also situations in which we attempt to reverse time in the solution of a time dependent equation to determine the initial conditions. There are many applications of inverse problems in areas as diverse as geophysics, remote sensing, astronomy and medical imaging. Inverse problems are typically ill-posed in the sense that even if there is a unique solution, small changes in the data can lead to arbitrarily large changes in the solution. Since measurements are always noisy in practice, this puts severe limits on our ability to solve these inverse problems.
This book is a not a research monograph presenting new results, but rather a broad survey of the mathematical theory of inverse problems from the point of view of functional analysis and partial differential equations. The book does not delve into statistical issues involved in quantifying the uncertainty in an inverse solution, and the book does not go into any depth on computational techniques for the numerical solution of inverse problems. Readers interested in these aspects of the subject would be better served by other books (see the references below).
The book begins with four introductory chapters that introduce inverse problems, ill-posedness, discrete ill-posed inverse problems in linear algebra, and integral equations. Each of the remaining chapters discusses a particular class of inverse problems, including problems of integral geometry, inverse scattering, and inverse problems associated with hyperbolic, parabolic, and elliptic partial differential equations.
The author has taken several steps to try to make this book more accessible to readers. A summary of notation precedes the table of contents. A seventy page appendix summarizes required background in functional analysis and partial differential equations. Exercises are scattered throughout the text, and a second appendix includes some additional exercises. The chapter on discrete linear inverse problems provides a more intuitive introduction to ideas that appear in later chapters in functional analytic form.
Some other aspects of the book are not so helpful to the reader. There is an extensive bibliography and additional references that take up 40 pages. Unfortunately, many of the references in this bibliography are in Russian and do not appear to be available in translation. The index is very brief and not at all adequate for a book of this length.
This book should be of interest as a reference work for researchers in inverse problems. The book could also prove to be useful as the textbook for an advanced graduate class on the mathematical theory of inverse problems.
 J. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems. Springer, 2005.
 J. L. Muller and S. Siltanen. Linear and Nonlinear Inverse Problems with Practical Applications. SIAM, 2012.
 A. Tarantola. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, 2004.
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. With Richard C. Aster and Clifford H. Thurber, he is the author of Parameter Estimation and Inverse Problems.