Regularization properties of LSQR for linear discrete ill-posed problems in the multiple singular value case and best, near best and general low rank approximations* Article Swipe
For the large-scale linear discrete ill-posed problem min‖ Ax − b ‖ or Ax = b with b contaminated by white noise, the Golub–Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the conjugate gradient (CG) method applied to A T Ax = A T b , are most commonly used. They have intrinsic regularizing effects, where the iteration number k plays the role of regularization parameter. The long-standing fundamental question is: Can LSQR and CGLS find two-norm filtering best possible regularized solutions ? The author has given definitive answers to this question for severely and moderately ill-posed problems when the singular values of A are simple. This paper extends the results to the multiple singular value case, and studies the approximation accuracy of Krylov subspaces, the quality of low rank approximations generated by Golub–Kahan bidiagonalization and the convergence properties of Ritz values. For the two kinds of problems, we prove that LSQR finds two-norm filtering best possible regularized solutions at semi-convergence. Particularly, we consider some important and untouched issues on best, near best and general rank k approximations to A for the ill-posed problems with the singular values with α > 0, and the relationships between them and their nonzero singular values. Numerical experiments confirm our theory. The results on general rank k approximations and the properties of their nonzero singular values apply to several Krylov solvers, including LSQR, CGME, MINRES, MR-II, GMRES and RRGMRES.
Related Topics
Concepts
Mathematics
Singular value
Regularization (linguistics)
Applied mathematics
Krylov subspace
Conjugate gradient method
Rank (graph theory)
Well-posed problem
Norm (philosophy)
Rate of convergence
Singular value decomposition
Linear subspace
Matrix norm
Pure mathematics
Linear system
Mathematical analysis
Combinatorics
Mathematical optimization
Algorithm
Eigenvalues and eigenvectors
Law
Electrical engineering
Engineering
Political science
Artificial intelligence
Computer science
Physics
Channel (broadcasting)
Quantum mechanics
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1088/1361-6420/ab9c45
- OA Status
- green
- Cited By
- 3
- References
- 55
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3034522293
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W3034522293Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1088/1361-6420/ab9c45Digital Object Identifier
- Title
-
Regularization properties of LSQR for linear discrete ill-posed problems in the multiple singular value case and best, near best and general low rank approximations*Work title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2020Year of publication
- Publication date
-
2020-06-12Full publication date if available
- Authors
-
Zhongxiao JiaList of authors in order
- Landing page
-
https://doi.org/10.1088/1361-6420/ab9c45Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/abs/2003.09259Direct OA link when available
- Concepts
-
Mathematics, Singular value, Regularization (linguistics), Applied mathematics, Krylov subspace, Conjugate gradient method, Rank (graph theory), Well-posed problem, Norm (philosophy), Rate of convergence, Singular value decomposition, Linear subspace, Matrix norm, Pure mathematics, Linear system, Mathematical analysis, Combinatorics, Mathematical optimization, Algorithm, Eigenvalues and eigenvectors, Law, Electrical engineering, Engineering, Political science, Artificial intelligence, Computer science, Physics, Channel (broadcasting), Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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3Total citation count in OpenAlex
- Citations by year (recent)
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2021: 1, 2020: 2Per-year citation counts (last 5 years)
- References (count)
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55Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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| referenced_works | https://openalex.org/W1972510094, https://openalex.org/W4302564868, https://openalex.org/W637574335, https://openalex.org/W1977271024, https://openalex.org/W1985318908, https://openalex.org/W2313084557, https://openalex.org/W2015071882, https://openalex.org/W1983886273, https://openalex.org/W2964137678, https://openalex.org/W1931725016, https://openalex.org/W625895441, https://openalex.org/W1491683340, https://openalex.org/W1986721151, https://openalex.org/W2073835183, https://openalex.org/W1594234351, https://openalex.org/W1998762105, https://openalex.org/W3217247658, https://openalex.org/W2316564661, https://openalex.org/W2611081798, https://openalex.org/W2067461769, https://openalex.org/W1488477687, https://openalex.org/W2964223338, https://openalex.org/W2314586021, https://openalex.org/W1975650588, https://openalex.org/W2803610866, https://openalex.org/W3008416987, https://openalex.org/W1531455566, https://openalex.org/W2483741779, https://openalex.org/W2069740856, https://openalex.org/W4293409666, https://openalex.org/W613352991, https://openalex.org/W4298264214, https://openalex.org/W2086784326, https://openalex.org/W2123944402, https://openalex.org/W2097897435, https://openalex.org/W2068403935, https://openalex.org/W2964350706, https://openalex.org/W2043394915, https://openalex.org/W4242722506, https://openalex.org/W2061813658, https://openalex.org/W2013477725, https://openalex.org/W2114424556, https://openalex.org/W2140064412, https://openalex.org/W1741578612, https://openalex.org/W2010315317, https://openalex.org/W2899557605, https://openalex.org/W1847144746, https://openalex.org/W2058583833, https://openalex.org/W2212050418, https://openalex.org/W1542258000, https://openalex.org/W1581163326, https://openalex.org/W4285719527, https://openalex.org/W658559791, https://openalex.org/W2133560833, https://openalex.org/W1570089119 |
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