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The Resource 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

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

Label
Variational source conditions, quadratic inverse problems, sparsity promoting regularization : new results in modern theory of inverse problems and an application in laser optics
Title
Variational source conditions, quadratic inverse problems, sparsity promoting regularization
Title remainder
new results in modern theory of inverse problems and an application in laser optics
Statement of responsibility
Jens Flemming
Creator
Author
Subject
Language
eng
Summary
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.--
Member of
Assigning source
Provided by publisher
Cataloging source
N$T
http://library.link/vocab/creatorName
Flemming, Jens
Dewey number
515/.357
Index
index present
LC call number
QA371
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
Series statement
Frontiers in mathematics
http://library.link/vocab/subjectName
Inverse problems (Differential equations)
Label
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
Instantiates
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Intro; Preface; Contents; Part I Variational Source Conditions; 1 Inverse Problems, Ill-Posedness, Regularization; 1.1 Setting; 1.2 Ill-Posedness; 1.2.1 Global Definitions by Hadamard and Nashed; 1.2.2 Local Definitions by Hofmann and Ivanov; 1.2.3 Interrelations; 1.2.4 Nashed's Definition in Case of Uncomplemented Null Spaces; 1.3 Tikhonov Regularization; 2 Variational Source Conditions Yield Convergence Rates; 2.1 Evolution of Variational Source Conditions; 2.2 Convergence Rates; 3 Existence of Variational Source Conditions; 3.1 Main Theorem; 3.2 Special Cases
  • 3.2.1 Linear Equations in Hilbert Spaces3.2.2 Bregman Distance in Banach Spaces; 3.2.3 Vanishing Error Functional; Part II Quadratic Inverse Problems; 4 What Are Quadratic Inverse Problems?; 4.1 Definition and Basic Properties; 4.2 Examples; 4.2.1 Autoconvolutions; 4.2.1.1 Autoconvolution of Functions with Uniformly Bounded Support; 4.2.1.2 Truncated Autoconvolution of Functions with Uniformly Bounded Support; 4.2.1.3 Autoconvolution of Periodic Functions; 4.2.2 Kernel-Based Autoconvolution in Laser Optics; 4.2.2.1 Ultra-Short Laser Pulses; 4.2.2.2 SD-SPIDER Method
  • 4.2.2.3 The Inverse Problem4.2.3 Schlieren Tomography; 4.3 Local Versus Global Ill-Posedness; 4.4 Geometric Properties of Quadratic Mappings' Ranges; 4.5 Literature on Quadratic Mappings; 5 Tikhonov Regularization; 6 Regularization by Decomposition; 6.1 Quadratic Isometries; 6.2 Decomposition of Quadratic Mappings; 6.3 Inversion of Quadratic Isometries; 6.4 A Regularization Method; 6.5 Numerical Example; 7 Variational Source Conditions; 7.1 About Variational Source Conditions; 7.2 Nonlinearity Conditions; 7.3 Classical Source Conditions; 7.4 Variational Source Conditions Are the Right Tool
  • 7.5 Sparsity Yields Variational Source ConditionsPart III Sparsity Promoting Regularization; 8 Aren't All Questions Answered?; 9 Sparsity and 1-Regularization; 9.1 Sparse Signals; 9.2 1-Regularization; 9.3 Other Sparsity Promoting Regularization Methods; 9.4 Examples; 9.4.1 Denoising; 9.4.2 Bidiagonal Operator; 9.4.3 Simple Integration and Haar Wavelets; 9.4.4 Simple Integration and Fourier Basis; 10 Ill-Posedness in the 1-Setting; 11 Convergence Rates; 11.1 Results in the Literature; 11.2 Classical Techniques Do Not Work; 11.3 Smooth Bases; 11.4 Non-smooth Bases
  • 11.5 Convergence Rates Without Source-Type Assumptions11.6 Convergence Rates Without Injectivity-Type Assumptions; 11.6.1 Distance to Norm Minimizing Solutions; 11.6.2 Sparse Solutions; 11.6.3 Sparse Unique Norm Minimizing Solution; 11.6.4 Non-sparse Solutions; 11.6.5 Examples; A Topology, Functional Analysis, Convex Analysis; A.1 Topological Spaces and Nets; A.2 Reflexivity, Weak and Weak* Topologies; A.3 Subdifferentials and Bregman Distances; B Verification of Assumption 11.13 for Example 11.18; References; Index
Dimensions
unknown
Extent
1 online resource.
File format
unknown
Form of item
online
Isbn
9783319952635
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
Label
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
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Intro; Preface; Contents; Part I Variational Source Conditions; 1 Inverse Problems, Ill-Posedness, Regularization; 1.1 Setting; 1.2 Ill-Posedness; 1.2.1 Global Definitions by Hadamard and Nashed; 1.2.2 Local Definitions by Hofmann and Ivanov; 1.2.3 Interrelations; 1.2.4 Nashed's Definition in Case of Uncomplemented Null Spaces; 1.3 Tikhonov Regularization; 2 Variational Source Conditions Yield Convergence Rates; 2.1 Evolution of Variational Source Conditions; 2.2 Convergence Rates; 3 Existence of Variational Source Conditions; 3.1 Main Theorem; 3.2 Special Cases
  • 3.2.1 Linear Equations in Hilbert Spaces3.2.2 Bregman Distance in Banach Spaces; 3.2.3 Vanishing Error Functional; Part II Quadratic Inverse Problems; 4 What Are Quadratic Inverse Problems?; 4.1 Definition and Basic Properties; 4.2 Examples; 4.2.1 Autoconvolutions; 4.2.1.1 Autoconvolution of Functions with Uniformly Bounded Support; 4.2.1.2 Truncated Autoconvolution of Functions with Uniformly Bounded Support; 4.2.1.3 Autoconvolution of Periodic Functions; 4.2.2 Kernel-Based Autoconvolution in Laser Optics; 4.2.2.1 Ultra-Short Laser Pulses; 4.2.2.2 SD-SPIDER Method
  • 4.2.2.3 The Inverse Problem4.2.3 Schlieren Tomography; 4.3 Local Versus Global Ill-Posedness; 4.4 Geometric Properties of Quadratic Mappings' Ranges; 4.5 Literature on Quadratic Mappings; 5 Tikhonov Regularization; 6 Regularization by Decomposition; 6.1 Quadratic Isometries; 6.2 Decomposition of Quadratic Mappings; 6.3 Inversion of Quadratic Isometries; 6.4 A Regularization Method; 6.5 Numerical Example; 7 Variational Source Conditions; 7.1 About Variational Source Conditions; 7.2 Nonlinearity Conditions; 7.3 Classical Source Conditions; 7.4 Variational Source Conditions Are the Right Tool
  • 7.5 Sparsity Yields Variational Source ConditionsPart III Sparsity Promoting Regularization; 8 Aren't All Questions Answered?; 9 Sparsity and 1-Regularization; 9.1 Sparse Signals; 9.2 1-Regularization; 9.3 Other Sparsity Promoting Regularization Methods; 9.4 Examples; 9.4.1 Denoising; 9.4.2 Bidiagonal Operator; 9.4.3 Simple Integration and Haar Wavelets; 9.4.4 Simple Integration and Fourier Basis; 10 Ill-Posedness in the 1-Setting; 11 Convergence Rates; 11.1 Results in the Literature; 11.2 Classical Techniques Do Not Work; 11.3 Smooth Bases; 11.4 Non-smooth Bases
  • 11.5 Convergence Rates Without Source-Type Assumptions11.6 Convergence Rates Without Injectivity-Type Assumptions; 11.6.1 Distance to Norm Minimizing Solutions; 11.6.2 Sparse Solutions; 11.6.3 Sparse Unique Norm Minimizing Solution; 11.6.4 Non-sparse Solutions; 11.6.5 Examples; A Topology, Functional Analysis, Convex Analysis; A.1 Topological Spaces and Nets; A.2 Reflexivity, Weak and Weak* Topologies; A.3 Subdifferentials and Bregman Distances; B Verification of Assumption 11.13 for Example 11.18; References; Index
Dimensions
unknown
Extent
1 online resource.
File format
unknown
Form of item
online
Isbn
9783319952635
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Quality assurance targets
not applicable
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote

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