Frame decompositions of bounded linear operators in Hilbert spaces with applications in tomography. (31st March 2021)

• Record Type: Journal Article
• Authors: Hubmer, Simon; Ramlau, Ronny
• Is Part Of: Inverse problems. Volume 37:Number 5(2021)
• Publication Date: 2021-03-31
• Subjects: frame decomposition -- singular-value decomposition -- inverse and ill-posed problems -- computerized tomography -- atmospheric tomography; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abe5b8
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.