Advertisement

Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

New Results in Modern Theory of Inverse Problems and an Application in Laser Optics

  • Jens Flemming

Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Variational Source Conditions

    1. Front Matter
      Pages 1-1
    2. Jens Flemming
      Pages 21-28
  3. Quadratic Inverse Problems

    1. Front Matter
      Pages 29-29
    2. Jens Flemming
      Pages 31-53
    3. Jens Flemming
      Pages 55-57
    4. Jens Flemming
      Pages 59-95
    5. Jens Flemming
      Pages 97-105
  4. Sparsity Promoting Regularization

    1. Front Matter
      Pages 107-107
    2. Jens Flemming
      Pages 109-110
    3. Jens Flemming
      Pages 111-123
    4. Jens Flemming
      Pages 125-127
    5. Jens Flemming
      Pages 129-164
  5. Back Matter
    Pages 165-182

About this book

Introduction

The book collects and contributes new results on the theory and practice of ill-posed inverse problems. 
Different notions of ill-posedness in Banach spaces for linear and nonlinear inverse problems are discussed not only in standard settings but also in situations up to now not covered by the literature. Especially, ill-posedness of linear operators with uncomplemented null spaces is examined.
Tools for convergence rate analysis of regularization methods are extended to a wider field of applicability. It is shown that the tool known as variational source condition always yields convergence rate results. 
A theory for nonlinear inverse problems with quadratic structure is developed as well as corresponding regularization methods. The new methods are applied to a difficult inverse problem from laser optics.
Sparsity promoting regularization is examined in detail from a Banach space point of view. Extensive convergence analysis reveals new insights into the behavior of Tikhonov-type regularization with sparsity enforcing penalty.

Keywords

inverse problem ill-posedness variational source condition quadratic inverse problem autoconvolution nonlinear inverse problem sparsity Tikhonov regularization source condition Banach space convergence rate

Authors and affiliations

  • Jens Flemming
    • 1
  1. 1.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-95264-2
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-95263-5
  • Online ISBN 978-3-319-95264-2
  • Series Print ISSN 1660-8046
  • Series Online ISSN 1660-8054
  • Buy this book on publisher's site
Industry Sectors
Electronics
Energy, Utilities & Environment