Linear Inverse Problems and Tikhonov Regularization

ISBN-13: 9780883851418 / Angielski / Twarda / 2016 / 336 str.

Mark Gockenbach

Opis

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.

Spis treści

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.