Improved Accuracy Estimation of the Tikhonov Method for Ill-Posed Optimization Problems in Hilbert Space

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 estima...

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Vydáno v:Computational mathematics and mathematical physics Ročník 63; číslo 4; s. 519 - 527
Hlavní autor: Kokurin, M. M.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Moscow Pleiades Publishing 01.04.2023
Springer Nature B.V
Témata:
Accuracy
Approximation
Computational Mathematics and Numerical Analysis
Estimates
General Numerical Methods
Hilbert space
Ill posed problems
Inequality
Mathematics
Mathematics and Statistics
Optimization
Regularization
Regularization methods
ill-posed optimization problem in Hilbert space
Tikhonov method
accuracy estimation
ISSN:0965-5425, 1555-6662
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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.
AbstractList 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.
Author Kokurin, M. M.
Author_xml – sequence: 1
  givenname: M. M.
  surname: Kokurin
  fullname: Kokurin, M. M.
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Cites_doi 10.1515/9783110255720
10.1088/0266-5611/26/2/025001
10.1134/S0965542517070090
10.3103/S1066369X1706007X
10.1515/jiip-2017-0004
10.1088/0266-5611/32/10/104008
10.1088/0266-5611/18/1/313
ContentType Journal Article
Copyright Pleiades Publishing, Ltd. 2023. ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2023, Vol. 63, No. 4, pp. 519–527. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2023, Vol. 63, No. 4, pp. 548–556.
Pleiades Publishing, Ltd. 2023.
Copyright_xml – notice: Pleiades Publishing, Ltd. 2023. ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2023, Vol. 63, No. 4, pp. 519–527. © Pleiades Publishing, Ltd., 2023. Russian Text © The Author(s), 2023, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2023, Vol. 63, No. 4, pp. 548–556.
– notice: Pleiades Publishing, Ltd. 2023.
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Tikhonov method
accuracy estimation
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KokurinM. Yu.Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov’s scheme for solving ill-posed optimization problemsRuss. Math.201761515910.3103/S1066369X1706007X1383.65065
KokurinM. Yu.Convergence rate estimates for Tikhonov’s scheme as applied to ill-posed nonconvex optimization problemsComput. Math. Math. Phys.20175711011110368141610.1134/S09655425170700901379.49012
SchusterT.KaltenbacherB.HofmannB.KazimierskiK.Regularization Methods in Banach Spaces2012BerlinDe Gruyter10.1515/97831102557201259.65087
M. Zhong and W. Wang, “A global minimization algorithm for Tikhonov functionals with p-convex (p ≥ 2) penalty terms in Banach spaces,” Inverse Probl. 32 (10), 104008 (2016).
KokurinM. Y.Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problemsJ. Inverse Ill-Posed Probl.201826463475383343210.1515/jiip-2017-00041398.65115
V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis: A University Course (RKhD, Moscow, 2011) [in Russian].
F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
S. W. Anzengruber and R. Ramlau, “Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators,” Inverse Probl. 26 (2), 025001 (2010).
TautenhahnU.On the method of Lavrentiev regularization for nonlinear ill-posed problemsInverse Probl.200218191207189359010.1088/0266-5611/18/1/3131005.65058
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References_xml – reference: V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis: A University Course (RKhD, Moscow, 2011) [in Russian].
– reference: F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
– reference: TautenhahnU.On the method of Lavrentiev regularization for nonlinear ill-posed problemsInverse Probl.200218191207189359010.1088/0266-5611/18/1/3131005.65058
– reference: BakushinskiiA. B.GoncharskiiA. V.Iterative Methods for Solving Ill-Posed Problems1989MoscowNauka0676.65050
– reference: S. W. Anzengruber and R. Ramlau, “Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators,” Inverse Probl. 26 (2), 025001 (2010).
– reference: KokurinM. Yu.Convergence rate estimates for Tikhonov’s scheme as applied to ill-posed nonconvex optimization problemsComput. Math. Math. Phys.20175711011110368141610.1134/S09655425170700901379.49012
– reference: KokurinM. Yu.Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov’s scheme for solving ill-posed optimization problemsRuss. Math.201761515910.3103/S1066369X1706007X1383.65065
– reference: KokurinM. Y.Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problemsJ. Inverse Ill-Posed Probl.201826463475383343210.1515/jiip-2017-00041398.65115
– reference: SchusterT.KaltenbacherB.HofmannB.KazimierskiK.Regularization Methods in Banach Spaces2012BerlinDe Gruyter10.1515/97831102557201259.65087
– reference: M. Zhong and W. Wang, “A global minimization algorithm for Tikhonov functionals with p-convex (p ≥ 2) penalty terms in Banach spaces,” Inverse Probl. 32 (10), 104008 (2016).
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SubjectTerms Accuracy
Approximation
Computational Mathematics and Numerical Analysis
Estimates
General Numerical Methods
Hilbert space
Ill posed problems
Inequality
Mathematics
Mathematics and Statistics
Optimization
Regularization
Regularization methods
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Title Improved Accuracy Estimation of the Tikhonov Method for Ill-Posed Optimization Problems in Hilbert Space
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