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.
Table of Contents
Excerpt: Chapter 2: Well-posed, ill-posed, and inverse problems (p. 15)
The purpose of this chapter is to explain the properties that a problem must have to be considered an inverse problem, and to study them in some detail. We are going to restrict ourselves to linear inverse problems defined on Hilbert spaces. Throughout this book, X and Y will denote Hilbert spaces and T : X → Y will denote a continuous linear operator. We wish to study the equation
T x = y, (2.1)
where y ∈ Y is given and x ∈ X is to be determined. It may be straightforward or difficult to solve accurately, depending on the properties of T. In this chapter, we describe the conditions that make (2.1) well-posed, ill-posed, or an inverse problem. An inverse problem is a special kind of ill-posed problem that is particularly difficult to solve, and such problems are the subject of this book.
About the Author
Mark Gockenbach received his PhD in Computational and Applied Mathematics from Rice University in 1994. He has held faculty positions at Indiana University (teaching in the ITM/MUCIA-Indiana University cooperative program in Malaysia for two years), the University of Michigan, and Rice University. He is now Professor and Chair of the Department of Mathematical Sciences at Michigan Technological University. Professor Gockenbach has won several awards for teaching, and he currently serves as a volunteer lecturer in the International Mathematical Union’s Volunteer Lecturer Program (VLP). As a VLP lecturer, he has taught master’s degree courses in Phnom Penh, Cambodia.
Professor Gockenbach’s research interests are primarily in inverse problems in partial differential equations. His previous books are Partial Differential Equations: Analytical and Numerical Methods (first edition 2002, second edition 2010) and Understanding and Implementing the Finite Element Method (2006), both published by the Society for Industrial and Applied Mathematics, and Finite-Dimensional Linear Algebra (2010), published by CRC Press.