Linear Inverse Problems and Tikhonov Regularization - Mark Gockenbach

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Résumé :
Inverse problems occur frequently in science and technology, whenever we need to infer causes from effects that we can measure. Mathematically, they are difficult problems because they are unstable: small bits of noise in the measurement can completely throw off the solution. Nevertheless, there are methods for finding good approximate solutions.

Linear Inverse Problems and Tikhonov Regularization examines one such method: Tikhonov regularization for linear inverse problems defined on Hilbert spaces. This is a clear example of the power of applying deep mathematical theory to solve practical problems. Beginning with a basic analysis of Tikhonov regularization, this book introduces the singular value expansion for compact operators, and uses it to explain why and how the method works. Tikhonov regularization with seminorms is also analyzed, which requires introducing densely defined unbounded operators and their basic properties. Some of the relevant background is included in appendices, making the book accessible to a wide range of readers....

Biographie:
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).

Sommaire:
Preface; 1. Introduction to inverse problems; 2. Well-posed, ill-posed, and inverse problems; 3. Tikhonov regularization; 4. Compact operators and the singular value expansion; 5. Tikhonov regularization with seminorms; Epilogue; A. Basic Hilbert space theory; B. Sobolev spaces; Bibliography; Index.