Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization

Beschreibung

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.
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.
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

Autor*in

Jens Flemming

Themen in »Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization«

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

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“The book presents a nice summary of many interesting recent results on Tikhonov regularization for ill-posed and inverse problems … . The book … contains various interesting refinements and improvements. It is written in a very clear style, the material is well organized, and there is an extensive bibliography with useful comments on the most relevant literature. Hence, it is a very welcome addition to the modern standard references on ill-posed and inverse problems.” (Bangti Jin, zbMATH 1530.65007, 2024)

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