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Applicable Analysis
An International Journal
Volume 46, 1992 - Issue 1-2
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Original Articles

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

Andreas Neubauer Institut für Mathernatik, Universität Linz, A-4040 Linz, Austria
Pages 59-72 | Received 28 Sep 1990, Published online: 10 May 2007

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Rebecca Rothermel, Wladimir Panfilenko, Prateek Sharma, Anne Wald, Thomas Schuster, Anne Jung & Stefan Diebels. (2022) A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals. Applied Mathematics in Science and Engineering 30:1, pages 141-165.
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M. Pricop-Jeckstadt. (2019) Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems. Inverse Problems in Science and Engineering 27:2, pages 205-236.
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U. Aßmann & A. Rösch. (2015) Regularization in Sobolev Spaces with Fractional Order. Numerical Functional Analysis and Optimization 36:3, pages 271-286.
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Jaan Janno & Ulrich Tautenhahn. (2003) On Lavrentiev Regularization for Ill-Posed Problems in Hilbert Scales. Numerical Functional Analysis and Optimization 24:5-6, pages 531-555.
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Articles from other publishers (48)

Chitra Mekoth, Santhosh George, P. Jidesh & Yeol Je Cho. (2022) Projection method for Fractional Lavrentiev Regularisation method in Hilbert scales. The Journal of Analysis 31:2, pages 1303-1333.
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Chitra MEKOTH, Santhosh GEORGE & Jidesh P. (2023) Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales. Hacettepe Journal of Mathematics and Statistics 52:3, pages 729-752.
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Chitra Mekoth, Santhosh George, P. Jidesh & Shobha M. Erappa. (2022) Finite dimensional realization of fractional Tikhonov regularization method in Hilbert scales. Partial Differential Equations in Applied Mathematics 5, pages 100246.
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Zhenwu Fu, Bo Han & Yong Chen. (2022) Levenberg–Marquardt method with general convex penalty for nonlinear inverse problems. Journal of Computational and Applied Mathematics 404, pages 113771.
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Chitra Mekoth, Santhosh George & P. Jidesh. (2021) Fractional Tikhonov regularization method in Hilbert scales. Applied Mathematics and Computation 392, pages 125701.
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Donghong Zhao, Yonghua Fan, Haoyu Liu & Yafeng Yang. (2021) An algorithm combining coordinate descent and split Bregman iterative for fractional image denoising model. Journal of Algorithms & Computational Technology 15, pages 174830262110311.
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Murilo Teixeira Silva, Reza Shahidi, Eric W. Gill & Weimin Huang. (2020) Nonlinear Extraction of Directional Ocean Wave Spectrum From Synthetic Bistatic High-Frequency Surface Wave Radar Data. IEEE Journal of Oceanic Engineering 45:3, pages 1004-1021.
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Bernd Hofmann & Christopher Hofmann. (2020) The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties. Mathematics 8:3, pages 331.
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Richard Nickl, Sara van de Geer & Sven Wang. (2020) Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems. SIAM/ASA Journal on Uncertainty Quantification 8:1, pages 374-413.
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Frank Werner & Bernd Hofmann. (2019) Convergence analysis of (statistical) inverse problems under conditional stability estimates. Inverse Problems 36:1, pages 015004.
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Tran Nhan Tam Quyen. (2019) Finite element analysis for identifying the reaction coefficient in PDE from boundary observations. Applied Numerical Mathematics 145, pages 297-314.
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Santhosh George & K. Kanagaraj. (2019) Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales. Computational Methods in Applied Mathematics 19:4, pages 765-778.
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Thorsten Hohage & Philip Miller. (2019) Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces. Inverse Problems 35:6, pages 065005.
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Enrique Otárola & Tran Nhan Tam Quyen. (2019) A reaction coefficient identification problem for fractional diffusion. Inverse Problems 35:4, pages 045010.
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H Egger & B Hofmann. (2018) Tikhonov regularization in Hilbert scales under conditional stability assumptions. Inverse Problems 34:11, pages 115015.
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Pallavi Mahale & Pradeep Kumar Dadsena. (2018) Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales. Computational Methods in Applied Mathematics 18:4, pages 687-702.
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Bernd Hofmann & Peter Mathé. (2018) Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems 34:1, pages 015007.
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Minghua ZhaoHui CaoXin ZhangZhenghao ShiPeng Li. Restoration of single image based on kernel estimation with L1-regularization method. Restoration of single image based on kernel estimation with L1-regularization method.
Murilo T. Silva, Eric W. Gill & Reza Shahidi. (2017) A new nonlinear approach to extraction of ocean wave spectra from bistatic Doppler HF-radar data. A new nonlinear approach to extraction of ocean wave spectra from bistatic Doppler HF-radar data.
Anne Wald & Thomas Schuster. (2017) Sequential subspace optimization for nonlinear inverse problems. Journal of Inverse and Ill-posed Problems 25:1, pages 99-117.
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Martin Burger, Barbara Kaltenbacher & Andreas Neubauer. 2015. Handbook of Mathematical Methods in Imaging. Handbook of Mathematical Methods in Imaging 431 470 .
Martin Burger, Barbara Kaltenbacher & Andreas Neubauer. 2020. Handbook of Mathematical Methods in Imaging. Handbook of Mathematical Methods in Imaging 1 37 .
V Naumova & S V Pereverzyev. (2013) Multi-penalty regularization with a component-wise penalization. Inverse Problems 29:7, pages 075002.
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Gisela L Mazzieri & Ruben D Spies. (2012) Regularization methods for ill-posed problems in multiple Hilbert scales. Inverse Problems 28:5, pages 055005.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2011) Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem. Numerische Mathematik 120:1, pages 45-77.
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Qinian Jin & Ulrich Tautenhahn. (2011) Implicit iteration methods in Hilbert scales under general smoothness conditions. Inverse Problems 27:4, pages 045012.
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Martin Burger, Barbara Kaltenbacher & Andreas Neubauer. 2011. Handbook of Mathematical Methods in Imaging. Handbook of Mathematical Methods in Imaging 345 384 .
S. Lu, S.V. Pereverzev, Y. Shao & U. Tautenhahn. (2010) On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales. Journal of Integral Equations and Applications 22:3.
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Jean-Michel Loubes & Carenne Ludeña. (2010) Penalized estimators for non linear inverse problems. ESAIM: Probability and Statistics 14, pages 173-191.
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Andreas Neubauer. (2010) Modified Tikhonov regularization for nonlinear ill-posed problems in Banach spaces. Journal of Integral Equations and Applications 22:2.
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Shuai Lu, Sergei V. Pereverzev, Yuanyuan Shao & Ulrich Tautenhahn. (2010) Discrepancy curves for multi-parameter regularization. Journal of Inverse and Ill-posed Problems 18:6.
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XiaoHong Chen & Demian Pouzo. (2009) On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing. Science in China Series A: Mathematics 52:6, pages 1157-1168.
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H Egger. (2008) Regularization of inverse problems with large noise. Journal of Physics: Conference Series 124, pages 012022.
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S. Kindermann, P. Mayer, H. Albrecher & H. Engl. (2008) Identification of the Local Speed Function in a Levy Model for Option Pricing. Journal of Integral Equations and Applications 20:2.
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Thorsten Hohage & Mihaela Pricop. (2008) Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging 2:2, pages 271-290.
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Herbert Egger. (2008) $\mathcal{Y}$-Scale Regularization. SIAM Journal on Numerical Analysis 46:1, pages 419-436.
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H. Egger. (2007) PreconditioningCGNE iteration for inverse problems. Numerical Linear Algebra with Applications 14:3, pages 183-196.
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Peter Mathé & Ulrich Tautenhahn. (2006) Interpolation in variable Hilbert scales with application to inverse problems. Inverse Problems 22:6, pages 2271-2297.
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Herbert Egger. (2006) Semi-iterative Regularization in Hilbert Scales. SIAM Journal on Numerical Analysis 44:1, pages 66-81.
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M T Nair, S V Pereverzev & U Tautenhahn. (2005) Regularization in Hilbert scales under general smoothing conditions. Inverse Problems 21:6, pages 1851-1869.
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Herbert Egger & Andreas Neubauer. (2005) Preconditioning Landweber iteration in Hilbert scales. Numerische Mathematik 101:4, pages 643-662.
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Ronny Ramlau. (2002) A steepest descent algorithm for the global minimization of the Tikhonov functional. Inverse Problems 18:2, pages 381-403.
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J Cheng & M Yamamoto. (2000) One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization . Inverse Problems 16:4, pages L31-L38.
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Jin Qi-nian. (2000) Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales. Inverse Problems 16:1, pages 187-197.
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Jin Qi-nian. (1999) Applications of the Modified Discrepancy Principle to Tikhonov Regularization of Nonlinear Ill-Posed Problems. SIAM Journal on Numerical Analysis 36:2, pages 475-490.
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Ulrich Tautenhahn. (1998) On a general regularization scheme for nonlinear ill-posed problems: II. Regularization in Hilbert scales. Inverse Problems 14:6, pages 1607-1616.
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Barbara Blaschke-Kaltenbacher & Heinz W. Engl. 1997. Inverse Problems in Medical Imaging and Nondestructive Testing. Inverse Problems in Medical Imaging and Nondestructive Testing 17 35 .
U Tautenhahn. (1994) Error estimates for regularized solutions of non-linear ill-posed problems. Inverse Problems 10:2, pages 485-500.
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