Regularization Methods in Banach Spaces

Thomas Schuster; Barbara Kaltenbacher; Bernd Hofmann; Kamil S. Kazimierski

Published by De Gruyter 2012

Regularization Methods in Banach Spaces

Thomas Schuster , Barbara Kaltenbacher , Bernd Hofmann and Kamil S. Kazimierski

Volume 10 in the series Radon Series on Computational and Applied Mathematics

https://doi.org/10.1515/9783110255720

Overview
Contents

About this book

Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods.

This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.

Author / Editor information

Thomas Schuster, Carl von Ossietzky Universität Oldenburg, Germany; Barbara Kaltenbacher, University of Stuttgart, Germany; Bernd Hofmann, Chemnitz University of Technology, Germany; Kamil S. Kazimierski, University of Bremen, Germany.

Topics

Frontmatter Requires Authentication Unlicensed Licensed Download PDF	i
Preface Requires Authentication Unlicensed Licensed Download PDF	vii
Contents Requires Authentication Unlicensed Licensed Download PDF	ix
Part I. Why to use Banach spaces in regularization theory? Requires Authentication Unlicensed Licensed Download PDF	1
Part II. Geometry and mathematical tools of Banach spaces Requires Authentication Unlicensed Licensed Download PDF	25
Part III. Tikhonov-type regularization Requires Authentication Unlicensed Licensed Download PDF	77
Part IV. Iterative regularization Requires Authentication Unlicensed Licensed Download PDF	141
Part V. The method of approximate inverse Requires Authentication Unlicensed Licensed Download PDF	233
Bibliography Requires Authentication Unlicensed Licensed Download PDF	265
Index Requires Authentication Unlicensed Licensed Download PDF	280

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Researchers, Lecturers, and Graduate Students in Mathematics; Academic Libraries
Pages
- Front matter: 11
- Main content: 283
Keywords: Banach Space; Iterative Method; Regularization Theory; Tikhonov Regularization

eBook

Published: July 30, 2012
ISBN: 9783110255720

Hardcover

Published: July 16, 2012
ISBN: 9783110255249