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 (Autor)

Buch | Softcover

XI, 182 Seiten

2018
Springer International Publishing (Verlag)
978-3-319-95263-5 (ISBN)

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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.

Inverse problems, ill-posedness, regularization.- Variational source conditions yield convergence rates.- Existence of variational source conditions.- What are quadratic inverse problems?.- Tikhonov regularization.- Regularization by decomposition.- Variational source conditions.- Aren't all questions answered?.- Sparsity and 1-regularization.- Ill-posedness in the l1-setting.- Convergence rates.

"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)

Erscheinungsdatum	20.09.2018
Reihe/Serie	Frontiers in Mathematics
Zusatzinfo	XI, 182 p. 35 illus.
Verlagsort	Cham
Sprache	englisch
Maße	168 x 240 mm
Gewicht	338 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	autoconvolution • Banach space • Convergence Rate • ill-posedness • inverse problem • Nonlinear inverse problem • quadratic inverse problem • source condition • sparsity • Tikhonov regularization • variational source condition
ISBN-10	3-319-95263-3 / 3319952633
ISBN-13	978-3-319-95263-5 / 9783319952635
Zustand	Neuware