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- 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; Index3.2.1 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 Method4.2.2.3 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 Tool7.5 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 BasesIncludes bibliographical references and index.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 CasesThe 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.--