Minimization of Tikhonov Functionals in  Banach Spaces

Bonesky, Thomas; Kazimierski, Kamil S.; Maaß, Peter; Schöpfer, Frank; Schuster, Thomas

doi:10.1155/2008/192679

Cited by 66 publications

(93 citation statements)

References 23 publications

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“…In order to show that the constant C Z in (3.18) is the same as in (3.13) we repeat the proof given in [16]: The assertion directly follows by substituting (3.13) for u = z − v and arbitrary v, z ∈ Z into the Bregman distance:…”

Section: Duality Mappings and Bregman Distance 55mentioning

confidence: 87%

Phase retrieval problems in x-ray physics

Homann

2015

Göttingen Series in X-Ray Physics

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Section: Duality Mappings and Bregman Distance 55mentioning

confidence: 87%

Phase retrieval problems in x-ray physics

Homann

2015

Göttingen Series in X-Ray Physics

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“…As a consequence, for the numerical treatment of direct regularization methods in combination with discretization approaches, iterative procedures are also required. In this context, the details have been omitted, and the reader is referred to the monographs [25,35,107,116] and to the sample [16,31,69,72,101] of papers from a comprehensive set of publications on numerical approaches.…”

Section: Theory Of Direct Regularization Methodsmentioning

confidence: 99%

“…For recent progress of regularization theory applied to linear ill-posed problems, please refer to the papers [16,22,31,57,61,80,85,86,90]. It is well known that inverse problems aimed at the identification of parameter functions in differential equations or boundary conditions from observations of state variables are in general nonlinear even if the differential equations are linear.…”

Section: Introductionmentioning

confidence: 99%

Regularization Methods for Ill-Posed Problems

Cheng

Hofmann

2014

Handbook of Mathematical Methods in Imaging

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In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

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“…Bonesky et al [34] have investigated powers of Banach space norm, to which also inverse scale space methods and their analysis were generalized in [51]. Motivated by the examples mentioned above and the modeling of textures, Meyer [134] proposed to use the dual norm of BV as a fidelity, i.e.…”

Section: Other Fidelitiesmentioning

confidence: 99%

A Guide to the TV Zoo

Burger

Osher²

2013

Level Set and PDE Based Reconstruction Methods in Imaging

118

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Total variation methods and similar approaches based on regularizations with`1-type norms (and seminorms) have become a very popular tool in image processing and inverse problems due to peculiar features that cannot be realized with smooth regularizations. In particular total variation techniques had particular success due to their ability to realize cartoon-type reconstructions with sharp edges. Due to an explosion of new developments in this field within the last decade it is a difficult task to keep an overview of the major results in analysis, the computational schemes, and the application fields. With these lectures we attempt to provide such an overview, of course biased by our major lines of research.We are focusing on the basic analysis of total variation methods and the extension of the original ROF-denoising model due various application fields. Furthermore we provide a brief discussion of state-of-the art computational methods and give an outlook to applications in different disciplines.

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Minimization of Tikhonov Functionals in Banach Spaces

Cited by 66 publications

References 23 publications

Phase retrieval problems in x-ray physics

Phase retrieval problems in x-ray physics

Regularization Methods for Ill-Posed Problems

A Guide to the TV Zoo

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