Email this article Implicit iterative schemes based on singular decomposition and regularizing algorithms
- Authors: Zhdanov A.I1
-
Affiliations:
- Samara State Technical University
- Issue: Vol 22, No 3 (2018)
- Pages: 549-556
- Section: Articles
- URL: https://journal-vniispk.ru/1991-8615/article/view/20607
- DOI: https://doi.org/10.14498/vsgtu1592
- ID: 20607
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Abstract
A new version of the simple iterations implicit method based on the singular value decomposition is proposed. It is shown that this variant of the simple iterations implicit method can significantly improve the computational stability of the algorithm and at the same time provides a high rate of its convergence. The application of the simple iterations implicit method based on the singular value decomposition for the development of iterative regularization algorithms is considered. The proposed algorithms can be effectively used to solve a wide class of ill-posed and ill-conditioned computational problems.
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##article.viewOnOriginalSite##About the authors
Alexander I Zhdanov
Samara State Technical University
Email: zhdanovaleksan@yandex.ru
Dr. Phys. & Math. Sci., Professor; Head of Department; Dept. of Higher Mathematics & Applied Computer Science 244, Molodogvardeyskaya st., Samara, 443100, Russian Federation
References
- Vainikko G. M., Veretennikov A. Yu. Iteration Procedures in Ill-Posed Problems. Moscow, Nauka, 1986, 186 pp. (In Russian)
- Bakushinsky A. B., Goncharsky A. V. Iterative methods for solving ill-posed problems. Moscow, Nauka, 1986, 186 pp. (In Russian)
- Matysik O. V. Explicit and implicit iteration procedures of solving ill-posed problems. Brest, Brest State Univ., 1986, 186 pp. (In Russian)
- Donatelli M. On nondecreasing sequences of regularization parameters for nonstationary iterated Tikhonov, Numer. Algor., 2012, vol. 60, no. 4, pp. 651-668. doi: 10.1007/s11075-012-9593-7.
- Landi G., Loli Picolomini E., Tomba I. A stopping criterion for iterative regularization methods, Appl. Numer. Math., 2016, vol. 106, pp. 53-68. doi: 10.1016/j.apnum.2016.03.006.
- Buccini A., Donatelli M., Reichel L. Iterated Tikhonov regularization with a general penalty term, Numer Linear Algebra Appl., 2017, vol. 24, no. 4, e2089. doi: 10.1002/nla.2089.
- Golub G. H., Van Loan C. F. Matrix computations, Johns Hopkins Series in the Mathematical Sciences, vol. 3. Baltimore, etc., The Johns Hopkins University Press., 1989, xix+642 pp.
- Björk Å. Numerical Methods in Matrix Computations, Texts in Applied Mathematics, vol. 59. Cham, Springer, 2015, xvi+800 pp. doi: 10.1007/978-3-319-05089-8.
- Malyshev A. N. Introduction to numerical linear algebra (with an application of algorithms on FORTRAN). Novosibirsk, Nauka, 1991, 229 pp. (In Russian)
- Hansen P. C. Rank-deficient and discrete ill-posed problems. Numerical aspects of linear inversion, SIAM Monographs on Mathematical Modeling and Computation, vol. 4. Philadelphia, PA, SIAM, Society for Industrial and Applied Mathematics, 1997, 247 pp.
- Yamazaki I., Tomov S., Dongarra J. Sampling Algorithms to Update Truncated SVD, In: IEEE International Conference on Big Data (Big Data), 2017. doi: 10.1109/BigData.2017.8257997.
- Gates M., Tomov S., Dongarra J. Accelerating the SVD Two StageBidiagonal Reduction and Divide and Conquer Using GPUs, Parallel Computing, 2018, vol. 74, pp. 3-18. doi: 10.1016/j.parco.2017.10.004.
- Kabir K., Haidar A., Tomov S., Bouteiller A., Dongarra J. A Framework for Out of Memory SVD Algorithms, In: High Performance Computing, ISC 2017. Lecture Notes in Computer Science, vol. 10266; eds. J. Kunkel, R. Yokota, P. Balaji, D. Keyes. Cham, Springer, pp. 158-178. doi: 10.1007/978-3-319-58667-0_9.
- Dong T., Haidar A., Tomov S., Dongarra J. Optimizing the SVD Bidiagonalization Process for a Batch of Small Matrices, Procedia Computer Science, 2017, vol. 108, pp. 1008-1018. doi: 10.1016/j.procs.2017.05.237.
- Haidar A., Kabir K., Fayad D., Tomov S., Dongarra J. Out Of Memory SVD Solver for Big Data, In: 2017 IEEE High Performance Extreme Computing Conference (HPEC), 2017. doi: 10.1109/HPEC.2017.8091029.
- Verzhbitskij V. M. Numerical Methods (Linear Algebra and Nonlinear Equations). Moscow, Oniks 21 Century Publishing House, 2005, 432 pp. (In Russian)
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