Injectivity and \(\text{weak}^\star\)-to-weak continuity suffice for convergence rates in \(\ell^{1}\)-regularization

DOI10.1515/JIIP-2017-0008zbMath1382.65159arXiv1701.03460OpenAlexW2963398839WikidataQ122949476 ScholiaQ122949476MaRDI QIDQ682035

Jens Flemming, Daniel Gerth

Publication date: 13 February 2018

Published in: Journal of Inverse and Ill-Posed Problems (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1701.03460

zbMATH Keywords

convergence ratesTikhonov regularizationlinear ill-posed problemsource conditionsparsity promoting regularizationvariational source condition

Mathematics Subject Classification ID

Numerical solutions to equations with linear operators (65J10) Numerical solutions of ill-posed problems in abstract spaces; regularization (65J20) Linear operators and ill-posed problems, regularization (47A52)

Related Items (12)

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations ⋮ Oversmoothing regularization with \(\ell^1\)-penalty term ⋮ On ℓ 1 -Regularization Under Continuity of the Forward Operator in Weaker Topologies ⋮ Convergence rates of a dual gradient method for constrained linear ill-posed problems ⋮ Penalty-based smoothness conditions in convex variational regularization ⋮ Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces ⋮ Existence of variational source conditions for nonlinear inverse problems in Banach spaces ⋮ Maximal spaces for approximation rates in \(\ell^1\)-regularization ⋮ Modern regularization methods for inverse problems ⋮ Variational Source Conditions in $L^p$-spaces ⋮ A new interpretation of (Tikhonov) regularization ⋮ Deep Neural Networks for Inverse Problems with Pseudodifferential Operators: An Application to Limited-Angle Tomography

Cites Work

• Unnamed Item
• Unnamed Item
• A unified approach to convergence rates for \(\ell^{1}\)-regularization and lacking sparsity
• Regularization methods in Banach spaces.
• Non-convex sparse regularisation
• Well-posedness and convergence rates for sparse regularization with sublinear \(l^q\) penalty term
• Convergence rates for ${{\ell }}^{1}$-regularization without injectivity-type assumptions
• Necessary and sufficient conditions for linear convergence of ℓ1-regularization
• Parameter choice in Banach space regularization under variational inequalities
• Regularization with non-convex separable constraints
• Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
• An Introduction to Banach Space Theory
• An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
• Convergence rates inℓ¹-regularization if the sparsity assumption fails
• Convergence rates inℓ¹-regularization when the basis is not smooth enough
• On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization
• Theory of operator algebras I.

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