Several mathematical problems in science, engineering, and technology are inverse problems. For example, inverse problem theory is often used in heat and mass transfer, imaging, hydrology, oceanography, and so on. In general, inverse problems are ill-posed, that is, the condition of stability is often violated. Several books cover the theoretical and numerical aspects of both ordinary and partial differential equations (ODEs and PDEs), whereas only a few books focus on the theoretical and numerical study of inverse problems. This book is an important contribution to the theory of inverse problems. It gives a complete picture of inverse problems and their applications.

The book begins with a nice introduction to the ill-posedness of inverse problems, along with some examples. Chapter 2 presents some basics of functional analysis, singular value decomposition, Tikhonov regularization, and Morozov’s discrepancy principle--concepts that are used in later chapters. Chapter 3 studies inverse source problems for heat and wave equations, and backward parabolic problems. It also discusses computational issues pertaining to inverse problems, along with some numerical examples.

Chapter 4 deals with inverse problems for second-order hyperbolic PDEs, including the existence of a local solution of the inverse problem, global stability, and uniqueness. One-dimensional inverse problems for electrodynamic equations are studied in chapter 5. Chapters 6 and 7 discuss inverse problems for parabolic and elliptic PDEs, respectively. Inverse problems for stationary transport equations are studied in chapter 8, and chapter 9 deals with inverse kinematic problems.

The book includes several mathematical theorems and proofs and some numerical examples. It begins with the basics of inverse problems and provides up-to-date information, including the latest developments. It is a good research monograph for people working on inverse problems and related issues; a useful state-of-the-art reference guide for researchers and students; and a fine textbook for graduate students in mathematics and engineering.