The field of inverse problems has experienced explosive growth in the last few decades. This is due in part to the importance of applications, like biomedical and seismic imaging, that require the practical solution of inverse problems. It is also due to the recent development of powerful computers and fast, reliable numerical methods with which to carry out the solution process. This monograph will provide the reader with a working understanding of these numerical methods. The intended audience includes graduate students and researchers in applied mathematics, engineering, and the physical sciences who may encounter inverse problems in their work.

Inverse problems typically involve the estimation of certain quantities based on indirect measurements of these quantities. For example, seismic exploration yields measurements of vibrations recorded on the earth's surface. These measurements are only indirectly related to the subsurface geological formations that are to be determined. The estimation process is often ill-posed in the sense that noise in the data may give rise to significant errors in the estimate. Techniques known as regularization methods have been developed to deal with this ill-posedness.

The first four chapters contain background material related to inverse problems, regularization, and numerical solution techniques. Chapter 1 provides an informal overview of a variety of regularization methods. Chapter 2 is guided by the philosophy that sensible numerical solutions to discretized problems follow from a thorough understanding of the underlying continuous problems. This chapter contains relevant functional analysis and infinite-dimensional optimization theory. Chapter 3 contains a review of relevant numerical optimization methods. Chapter 4 presents statistics material that pertains to inverse problems. This includes topics like maximum likelihood estimation and Bayesian estimation.