Although many problems in applied mathematics proceed directly from a description of a physical system to predictions of observable properties of the system, in inverse problems we begin with observations of a system and attempt to infer properties. These inverse problems are typically ill-posed, in the sense that the solution to the inverse problem can fail to exist or be extremely sensitive to small errors in the observations. Since in practice all measurements incorporate noise at some level, a naive solution of an inverse problem with noisy data is typically a meaningless solution. Such ill-posed inverse problems arise in many areas of science and engineering, including geophysical inversion, medical imaging, and inverse problems in heat transfer.
Inverse problems have been addressed in a variety of ways. One important line of research considers inverse problems from the point of view of regularization procedures that turn an ill-posed inverse problem into a sequence of well posed problems whose solutions converge to the solution of the original inverse problem in the limit as the noise is reduced to 0. Another line of research focuses on the recovery of coefficients in PDE boundary value problems and considers conditions under which these coefficients can be stably recovered.
The first half of this book discusses inverse problems in which the forward operator is a linear mapping between suitable spaces of functions. The authors approach these linear inverse problems by discretizing them to produce linear systems of equations and then regularizing the solution. Image deblurring, X-ray tomography, and backward heat propagation are used as guiding examples throughout these chapters.
The authors discuss strategies for regularizing these discretized problems including Tikhonov regularization and sparsity regularization. The strategy of discretizing an inverse problem to produce a linear system of equations and then regularizing the solution of the resulting ill-conditioned system of equations is a very general strategy that can be applied to any linear inverse problem. The authors have presented this material with a minimum of required mathematical background. It should be accessible to students majoring in mathematics, science, or engineering at the advanced undergraduate level.
The same general strategy can be applied to nonlinear inverse problems, with iterative methods used to optimize the solution of the discretized and regularized problem. However, the resulting nonlinear optimization problems may be nonconvex and suffer from local minima. Furthermore, there are significant theoretical difficulties in proving that this kind of iterative nonlinear regularization scheme converges in the limit as noise goes to 0.
An alternative approach to nonlinear inverse problems is to directly construct a nonlinear map from the data to the unknown parameters. The direct approach typically requires careful analysis followed by the numerical solution of a partial differential equation to obtain the inverse solution. This solution methodology is highly dependent on the particular inverse problem. The second half of the book focuses on one example nonlinear inverse problem, electrical impedance tomography. The authors present the D-bar method for the solution of the EIT problem. The material in this second half of the book is at a considerably more advanced mathematical level, though it should generally be accessible to graduate students in applied mathematics.
This book has clearly been written for use as a textbook. The authors have provided appendices that discuss important background in mathematical analysis and iterative methods for the solution of linear systems of equations. The authors have also included numerous exercises and suggestions for student projects. There is a companion web site that includes MATLAB code for many of the examples.
The first half of this textbook will be of interest to instructors who are teaching an introductory course in inverse problems that focuses on the regularization approach to linear inverse problems. The second half of the book, perhaps with some additional readings, would be suitable for a more advanced graduate course on direct methods for the solution of nonlinear inverse problems.
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.