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Applicable Analysis
An International Journal
Volume 93, 2014 - Issue 7
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Articles

Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

Stephan W. AnzengruberDepartment of Mathematics, Chemnitz University of Technology, Chemnitz09107, Germany.
,
Bernd HofmannDepartment of Mathematics, Chemnitz University of Technology, Chemnitz09107, Germany.Correspondence[email protected]
&
Peter MathéWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, Berlin10117, Germany.
Pages 1382-1400 | Received 06 Nov 2012, Accepted 06 Aug 2013, Published online: 03 Sep 2013

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Lihua Xu, Li Li, Wei Wang & Yiping Gao. (2020) CT image reconstruction algorithms based on the Hanke Raus parameter choice rule. Inverse Problems in Science and Engineering 28:1, pages 87-103.
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Wei Wang, Stephan W. Anzengruber, Ronny Ramlau & Bo Han. (2015) A global minimization algorithm for Tikhonov functionals with sparsity constraints. Applicable Analysis 94:3, pages 580-611.
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Qin Huang, Rongfang Gong, Qinian Jin & Ye Zhang. (2023) A Tikhonov regularization method for Cauchy problem based on a new relaxation model. Nonlinear Analysis: Real World Applications 74, pages 103935.
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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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Kuan Li, Chun Huang & Ziyang Yuan. (2021) Error Estimations for Total Variation Type Regularization. Mathematics 9:12, pages 1373.
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Stefan Gombots, Jonathan Nowak & Manfred Kaltenbacher. (2021) Sound source localization – state of the art and new inverse schemeSchallquellenlokalisation: state of the art und neues inverses Verfahren. e & i Elektrotechnik und Informationstechnik 138:3, pages 229-243.
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Erdem Altuntac. (2020) Choice of the parameters in a primal-dual algorithm for Bregman iterated variational regularization. Numerical Algorithms 86:2, pages 729-759.
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De-Han Chen & Irwin Yousept. (2021) Variational Source Conditions in $L^p$-spaces. SIAM Journal on Mathematical Analysis 53:3, pages 2863-2889.
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Stefan Gombots, Manfred Kaltenbacher & Barbara Kaltenbacher. (2021) Capabilities of inverse scheme for acoustic source localization at low frequencies. Acta Acustica 5, pages 44.
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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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Cong Shi, Claus Ropers & Thorsten Hohage. (2020) Density matrix reconstructions in ultrafast transmission electron microscopy: uniqueness, stability, and convergence rates. Inverse Problems 36:2, pages 025005.
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Frederic Weidling, Benjamin Sprung & Thorsten Hohage. (2020) Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces. SIAM Journal on Numerical Analysis 58:1, pages 21-47.
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Thorsten Hohage, Benjamin Sprung & Frederic Weidling. 2020. Nanoscale Photonic Imaging. Nanoscale Photonic Imaging 145 164 .
Leon Bungert & Martin Burger. (2019) Solution paths of variational regularization methods for inverse problems. Inverse Problems 35:10, pages 105012.
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Wei Wang, Shuai Lu, Bernd Hofmann & Jin Cheng. (2019) Tikhonov regularization with ℓ 0 -term complementing a~convex penalty: ℓ 1 -convergence under sparsity constraints . Journal of Inverse and Ill-posed Problems 27:4, pages 575-590.
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Bernd Hofmann, Stefan Kindermann & Peter Mathé. (2019) Penalty-based smoothness conditions in convex variational regularization. Journal of Inverse and Ill-posed Problems 27:2, pages 283-300.
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De-Han Chen & Irwin Yousept. (2019) Variational source condition for ill-posed backward nonlinear Maxwell’s equations. Inverse Problems 35:2, pages 025001.
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Erik Burman, Peter Hansbo & Mats G Larson. (2018) Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems 34:3, pages 035004.
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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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VINICIUS ALBANI, ADRIANO DE CEZARO & JORGE P. ZUBELLI. (2017) CONVEX REGULARIZATION OF LOCAL VOLATILITY ESTIMATION. International Journal of Theoretical and Applied Finance 20:01, pages 1750006.
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De-Han Chen, Bernd Hofmann & Jun Zou. (2017) Elastic-net regularization versus ℓ 1 -regularization for linear inverse problems with quasi-sparse solutions . Inverse Problems 33:1, pages 015004.
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Min Zhong & Wei Wang. (2016) A global minimization algorithm for Tikhonov functionals with $p-$convex ($p\,\geqslant \,2$) penalty terms in Banach spaces. Inverse Problems 32:10, pages 104008.
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Steven Bürger, Jens Flemming & Bernd Hofmann. (2016) On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation. Inverse Problems 32:10, pages 104006.
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Jens Flemming. (2016) Convergence rates for ${{\ell }}^{1}$-regularization without injectivity-type assumptions. Inverse Problems 32:9, pages 095001.
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Bernd Hofmann, Barbara Kaltenbacher & Elena Resmerita. (2016) Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems and Imaging 10:3, pages 741-764.
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Vinicius Albani, Adriano De Cezaro & Jorge P. Zubelli. (2016) On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems and Imaging 10:1, pages 1-25.
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Thorsten Hohage & Frederic Weidling. (2015) Verification of a variational source condition for acoustic inverse medium scattering problems. Inverse Problems 31:7, pages 075006.
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Bernd Hofmann. 2015. Mathematical and Computational Modeling. Mathematical and Computational Modeling 192 221 .
Oliver G. Ernst, Björn Sprungk & Hans-Jörg Starkloff. (2015) Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems. SIAM/ASA Journal on Uncertainty Quantification 3:1, pages 823-851.
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Jin Cheng & Bernd Hofmann. 2015. Handbook of Mathematical Methods in Imaging. Handbook of Mathematical Methods in Imaging 91 123 .
Jin Cheng & Bernd Hofmann. 2020. Handbook of Mathematical Methods in Imaging. Handbook of Mathematical Methods in Imaging 1 31 .
Oliver G. Ernst, Björn Sprungk & Hans-Jörg Starkloff. 2014. Extraction of Quantifiable Information from Complex Systems. Extraction of Quantifiable Information from Complex Systems 133 159 .
Vinicius Viana Luiz Albani, Adriano De Cezaro & Jorge P. Zubelli. (2014) On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy. SSRN Electronic Journal.
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Vinicius Viana Luiz Albani, Adriano De Cezaro & Jorge P. Zubelli. (2013) Convex Regularization of Local Volatility Estimation in a Discrete Setting. SSRN Electronic Journal.
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Vinicius Viana Luiz Albani & Jorge P. Zubelli. (2012) Online Local Volatility Calibration by Convex Regularization with Morozov's Principle and Convergence Rates. SSRN Electronic Journal.
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