Linear Inverse Problems and Tikhonov Regularization (Carus Mathematical Monographs)

Tikhonov regularization is the most popular general-purpose method for regularization, a mathematical technique to suppress the effect of noise in data, and uses much of the machinery of Hilbert space theory. This book develops the theory of Tikhonov regularization for a certain class of linear inverse problems which are defined on Hilbert spaces. To explain why and how Tikhonov regularization works, the singular value expansion for compact operators is introduced. Tikhonov regularization with seminorms is also analyzed and for this purpose, densely defined unbounded operators are addressed and their basic properties presented. In addition, the author provides readers with a quick but thorough review of Hilbert space theory and a brief introduction to weak derivatives and Sobolev spaces. Intended as an expository work for those interested in inverse problems and Tikhonov regularization, including graduates and researchers, the author presents the theory in an engaging and straightforward style.

Mark Gockenbach received his Ph.D. in Computational and Applied Mathematics from Rice University, Houston and has since held faculty positions at Indiana University, the University of Michigan and Rice University. He is now Professor and Chair of the Department of Mathematical Sciences at Michigan Technological University and has won several awards for teaching. He also serves as a volunteer lecturer in the International Mathematical Union's Volunteer Lecturer Program (VLP) where he has taught master's degree courses in Phnom Penh, Cambodia. He has published several books on inverse problems in partial differential equations, including Partial Differential Equations: Analytical and Numerical Methods (first edition 2002, second edition 2010) and Understanding and Implementing the Finite Element Method (2006).