Introduction to Inverse Problems for Differential Equation

Inverse problems associated with linear partial differential equation boundary value problems have applications in many areas of science and engineering. The forward problem is to determine a solution of the fully specified boundary value problem. In the inverse problem we are given a solution to the boundary value problem and work backwards to find the initial conditions, or the boundary conditions, or perhaps a source term in the equation. These problems are notorious for typically being ill-posed in the sense that the solution to the inverse problem can depend discontinuously on the solution to the forward problem. The subject is particularly difficult because it requires a considerable background in partial differential equations and functional analysis.

The first half of this book is a concise tutorial on inverse problems for linear partial differential equations. The authors begin with an introductory chapter that introduces the reader to inverse problems for partial differential equations and the concept of ill-posedness. This is followed by a chapter on the functional analysis of linear operators including adjoints, the Moore-Penrose pseudoinverse, the singular value decomposition, and Tikhonov regularization.

In the third chapter, these ideas are applied to some one-dimensional examples, including the inverse source problem for the heat equation, the inverse source problem for the wave equation, and the problem of determining the initial condition for the heat equation. In each case the inverse problem is formulated as a linear operator equation on suitable function spaces which can then be regularized by truncating the SVD or by Tikhonov regularization. The authors briefly discuss a numerical approach to solving these problems that uses the Galerkin finite element method to solve the forward and adjoint problems within a conjugate gradient scheme.

The second half of the book consists of chapters that summarize analytical results on particular inverse problems organized by the type of the partial differential equation. There are chapters on inverse problems for hyperbolic, parabolic, and elliptic equations as well as for transport equations. The material in this half of the book will mainly be of use as a reference for researchers who are already familiar with the subject.

Although the first half of this book is a well written and concise introduction to the analysis of inverse problems for partial differential equations, the presentation is dense and assumes a strong background in functional analysis and PDE. Furthermore, the book lacks exercises. Thus it might not be suitable for use as a textbook in a graduate level course. In comparison, Inverse Problems for Partial Differential Equations, 3rd ed. by Victor Isakov is a more accessible alternative.

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.