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
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The item 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 represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.This item is available to borrow from 1 library branch.
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The item 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 represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.
This item is available to borrow from 1 library branch.
 Summary
 The book collects and contributes new results on the theory and practice of illposed inverse problems. Different notions of illposedness 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, illposedness 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 Tikhonovtype regularization with sparsity enforcing penalty.
 Language
 eng
 Extent
 1 online resource.
 Contents

 Intro; Preface; Contents; Part I Variational Source Conditions; 1 Inverse Problems, IllPosedness, Regularization; 1.1 Setting; 1.2 IllPosedness; 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 KernelBased Autoconvolution in Laser Optics; 4.2.2.1 UltraShort Laser Pulses; 4.2.2.2 SDSPIDER Method
 4.2.2.3 The Inverse Problem4.2.3 Schlieren Tomography; 4.3 Local Versus Global IllPosedness; 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 1Regularization; 9.1 Sparse Signals; 9.2 1Regularization; 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 IllPosedness in the 1Setting; 11 Convergence Rates; 11.1 Results in the Literature; 11.2 Classical Techniques Do Not Work; 11.3 Smooth Bases; 11.4 Nonsmooth Bases
 11.5 Convergence Rates Without SourceType Assumptions11.6 Convergence Rates Without InjectivityType 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 Nonsparse 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
 Isbn
 9783319952635
 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
 Language
 eng
 Summary
 The book collects and contributes new results on the theory and practice of illposed inverse problems. Different notions of illposedness 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, illposedness 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 Tikhonovtype regularization with sparsity enforcing penalty.
 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
 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, IllPosedness, Regularization; 1.1 Setting; 1.2 IllPosedness; 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 KernelBased Autoconvolution in Laser Optics; 4.2.2.1 UltraShort Laser Pulses; 4.2.2.2 SDSPIDER Method
 4.2.2.3 The Inverse Problem4.2.3 Schlieren Tomography; 4.3 Local Versus Global IllPosedness; 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 1Regularization; 9.1 Sparse Signals; 9.2 1Regularization; 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 IllPosedness in the 1Setting; 11 Convergence Rates; 11.1 Results in the Literature; 11.2 Classical Techniques Do Not Work; 11.3 Smooth Bases; 11.4 Nonsmooth Bases
 11.5 Convergence Rates Without SourceType Assumptions11.6 Convergence Rates Without InjectivityType 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 Nonsparse 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
 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, IllPosedness, Regularization; 1.1 Setting; 1.2 IllPosedness; 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 KernelBased Autoconvolution in Laser Optics; 4.2.2.1 UltraShort Laser Pulses; 4.2.2.2 SDSPIDER Method
 4.2.2.3 The Inverse Problem4.2.3 Schlieren Tomography; 4.3 Local Versus Global IllPosedness; 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 1Regularization; 9.1 Sparse Signals; 9.2 1Regularization; 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 IllPosedness in the 1Setting; 11 Convergence Rates; 11.1 Results in the Literature; 11.2 Classical Techniques Do Not Work; 11.3 Smooth Bases; 11.4 Nonsmooth Bases
 11.5 Convergence Rates Without SourceType Assumptions11.6 Convergence Rates Without InjectivityType 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 Nonsparse 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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