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Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

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

  • Book
  • © 2018

Overview

Authors:
  1. Jens Flemming

    1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

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  • Presents recent research results in a consistent notation
  • Contributes to two very active fields of research
  • Shows that variational source condition always yields convergence rate results

Part of the book series: Frontiers in Mathematics (FM)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xi
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  2. Variational Source Conditions

    1. Front Matter

      Pages 1-1
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    2. Inverse Problems, Ill-Posedness, Regularization

      • Jens Flemming
      Pages 3-14

    3. Variational Source Conditions Yield Convergence Rates

      • Jens Flemming
      Pages 15-20

    4. Existence of Variational Source Conditions

      • Jens Flemming
      Pages 21-28

  3. Quadratic Inverse Problems

    1. Front Matter

      Pages 29-29
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    2. What Are Quadratic Inverse Problems?

      • Jens Flemming
      Pages 31-53

    3. Tikhonov Regularization

      • Jens Flemming
      Pages 55-57

    4. Regularization by Decomposition

      • Jens Flemming
      Pages 59-95

    5. Variational Source Conditions

      • Jens Flemming
      Pages 97-105

  4. Sparsity Promoting Regularization

    1. Front Matter

      Pages 107-107
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    2. Aren’t All Questions Answered?

      • Jens Flemming
      Pages 109-110

    3. Sparsity and ℓ 1-Regularization

      • Jens Flemming
      Pages 111-123

    4. Ill-Posedness in the ℓ 1-Setting

      • Jens Flemming
      Pages 125-127

    5. Convergence Rates

      • Jens Flemming
      Pages 129-164

  5. Back Matter

    Pages 165-182
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Keywords

About this book

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.

Authors and Affiliations

  • Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Jens Flemming

Bibliographic Information

  • Book Title: Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

  • Book Subtitle: New Results in Modern Theory of Inverse Problems and an Application in Laser Optics

  • Authors: Jens Flemming

  • Series Title: Frontiers in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-95264-2

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2018

  • Softcover ISBN: 978-3-319-95263-5Published: 18 September 2018

  • eBook ISBN: 978-3-319-95264-2Published: 08 September 2018

  • Series ISSN: 1660-8046

  • Series E-ISSN: 1660-8054

  • Edition Number: 1

  • Number of Pages: XI, 182

  • Number of Illustrations: 35 b/w illustrations

  • Topics: Numerical Analysis, Operator Theory

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