Regularization of linear ill-posed problems involving multiplication\n operators Article Swipe
Peter Mathé
,
M. Thamban Nair
,
Bernd Hofmann
·
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1908.05871
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1908.05871
We study regularization of ill-posed equations involving multiplication\noperators when the multiplier function is positive almost everywhere and zero\nis an accumulation point of the range of this function. Such equations\nnaturally arise from equations based on non-compact self-adjoint operators in\nHilbert space, after applying unitary transformations arising out of the\nspectral theorem. For classical regularization theory, when noisy observations\nare given and the noise is deterministic and bounded, then non-compactness of\nthe ill-posed equations is a minor issue. However, for statistical ill-posed\nequations with non-compact operators less is known if the data are blurred by\nwhite noise. We develop a regularization theory with emphasis on this case. In\nthis context, we highlight several aspects, in particular we discuss the\nintrinsic degree of ill-posedness in terms of rearrangements of the multiplier\nfunction. Moreover, we address the required modifications of classical\nregularization schemes in order to be used for non-compact statistical\nproblems, and we also introduce the concept of the effective ill-posedness of\nthe operator equation under white noise. This study is concluded with\nprototypical examples for such equations, as these are deconvolution equations\nand certain final value problems in evolution equations.\n
Related Topics
Concepts
Mathematics
Regularization (linguistics)
Bounded function
Multiplier (economics)
Well-posed problem
Hilbert space
Applied mathematics
Regularization perspectives on support vector machines
White noise
Mathematical analysis
Inverse problem
Computer science
Artificial intelligence
Statistics
Macroeconomics
Economics
Tikhonov regularization
Metadata
- Type
- preprint
- Landing Page
- http://arxiv.org/abs/1908.05871
- https://arxiv.org/pdf/1908.05871
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4288261843
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4288261843Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.1908.05871Digital Object Identifier
- Title
-
Regularization of linear ill-posed problems involving multiplication\n operatorsWork title
- Type
-
preprintOpenAlex work type
- Publication year
-
2019Year of publication
- Publication date
-
2019-08-16Full publication date if available
- Authors
-
Peter Mathé, M. Thamban Nair, Bernd HofmannList of authors in order
- Landing page
-
https://arxiv.org/abs/1908.05871Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/1908.05871Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/1908.05871Direct OA link when available
- Concepts
-
Mathematics, Regularization (linguistics), Bounded function, Multiplier (economics), Well-posed problem, Hilbert space, Applied mathematics, Regularization perspectives on support vector machines, White noise, Mathematical analysis, Inverse problem, Computer science, Artificial intelligence, Statistics, Macroeconomics, Economics, Tikhonov regularizationTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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