On Converse and Saturation Results for Tikhonov Regularization of Linear Ill-Posed Problems

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Previous article Next article On Converse and Saturation Results for Tikhonov Regularization of Linear Ill-Posed ProblemsAndreas NeubauerAndreas Neubauerhttps://doi.org/10.1137/S0036142993253928PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper we prove some new converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces.[1] Heinz Engl and , Helmut Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill‐posed problems, Appl. Numer. Math., 4 (1988), 395–417 10.1016/0168-9274(88)90017-7 89i:65060 CrossrefISIGoogle Scholar[2] Helmut Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill‐posed problems leading to optimal convergence rates, Math. Comp., 49 (1987), 0–0507–522, S5–S12 88k:65049 CrossrefISIGoogle Scholar[3] Google Scholar[4] C. Groetsch and , A. Neubauer, Regularization of ill‐posed problems: optimal parameter choice in finite dimensions, J. Approx. Theory, 58 (1989), 184–200 90j:65083 CrossrefISIGoogle Scholar[5] Martin Hanke, Accelerated Landweber iterations for the solution of ill‐posed equations, Numer. Math., 60 (1991), 341–373 92m:65077 CrossrefISIGoogle Scholar[6] J. King and , A. Neubauer, A variant of finite‐dimensional Tikhonov regularization with a posteriori parameter choice, Computing, 40 (1988), 91–109 90d:65107 CrossrefISIGoogle Scholar[7] A. Neubauer, Tikhonov‐regularization of ill‐posed linear operator equations on closed convex sets, J. Approx. Theory, 53 (1988), 304–320 89h:65094 CrossrefISIGoogle Scholar[8] T. Raus, The principle of the residual in the solution of ill‐posed problems, Tartu Riikl. Ül. Toimetised, (1984), 16–26 86d:65074 Google Scholar[9] T. Raus, On the residue principle for the solution of ill–posed problems with non–selfadjoint operator, (Russian), Uch. Zap. Tartu Gos. 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Math., Vol. 73, Birkhäuser, Basel, 1985, 234–243 89a:65093 Google ScholarKeywordslinear ill-posed problemsTikhonov regularizationconverse and saturation results Previous article Next article FiguresRelatedReferencesCited ByDetails On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problemsNumerische Mathematik, Vol. 32 | 14 September 2022 Cross Ref Regularization of linear ill-posed problems involving multiplication operatorsApplicable Analysis, Vol. 101, No. 2 | 28 April 2020 Cross Ref Estimating Solution Smoothness and Data Noise with Tikhonov RegularizationNumerical Functional Analysis and Optimization, Vol. 43, No. 1 | 28 November 2021 Cross Ref Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed ProblemsFoundations of Computational Mathematics, Vol. 31 | 17 August 2021 Cross Ref A new interpretation of (Tikhonov) regularizationInverse Problems, Vol. 37, No. 6 | 7 June 2021 Cross Ref A New Regularization Method for Linear 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Mathematics