Improved Accuracy Estimation of the Tikhonov Method for Ill-Posed Optimization Problems in Hilbert Space
- Authors: Kokurin M.M.1
-
Affiliations:
- Mari State University
- Issue: Vol 63, No 4 (2023)
- Pages: 548-556
- Section: ОБЩИЕ ЧИСЛЕННЫЕ МЕТОДЫ
- URL: https://journals.rcsi.science/0044-4669/article/view/136156
- DOI: https://doi.org/10.31857/S0044466923040117
- EDN: https://elibrary.ru/IPHNWS
- ID: 136156
Cite item
Abstract
The Tikhonov method is studied as applied to ill-posed problems of minimizing a smooth nonconvex functional. Assuming that the sought solution satisfies the source condition, an accuracy estimate for the Tikhonov method is obtained in terms of the regularization parameter. Previously, such an estimate was obtained only under the assumption that the functional is convex or under a structural condition imposed on its nonlinearity. Additionally, a new accuracy estimate for the Tikhonov method is obtained in the case of an approximately specified functional.
About the authors
M. M. Kokurin
Mari State University
Author for correspondence.
Email: comp_mat@ccas.ru
424000, Yoshkar-Ola, Russia
References
- Богачёв В.И., Смолянов О.Г. Действительный и функциональный анализ. М.-Ижевск: НИЦ “Регулярная и хаотическая динамика”, 2011.
- Васильев Ф.П. Методы решения экстремальных задач. М.: Наука, 1981.
- Бакушинский А.Б., Гончарский А.В. Итеративные методы решения некорректных задач. М.: Наука, 1989.
- Кокурин М.Ю. Необходимые и достаточные условия степенной сходимости приближений в схеме Тихонова для решения некорректных экстремальных задач // Известия вузов. Математика. 2017. № 6. С. 60–69.
- Tautenhahn U. On the method of Lavrentiev regularization for nonlinear ill–posed problems // Inverse Problems. 2002. V. 18. P. 191–207.
- Кокурин М.Ю. Оценки скорости сходимости в схеме Тихонова для решения некорректных невыпуклых экстремальных задач // Ж. вычисл. матем. и матем. физ. 2017. Т. 57. № 7. С. 1103–1112.
- Kokurin M.Y. Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill–posed nonconvex optimization problems // J. of Inverse and Ill–Posed Problems. 2018. V. 26. № 4. P. 463–475.
- Schuster T., Kaltenbacher B., Hofmann B., Kazimierski K. Regularization Methods in Banach Spaces // Radon Series on Computational and Applied Mathematics. 2012.
- Anzengruber S.W., Ramlau R. Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators // Inverse Problems. 2010. V. 26. № 2. 025001.
- Zhong M., Wang W. A global minimization algorithm for Tikhonov functionals with -convex () penalty terms in Banach spaces // Inverse Problems. 2016. V. 32. № 10. 104008.