A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators

By Bernd Hofmann, Barbara Kaltenbacher, C. Pöschl, Otmar Scherzer. 2007

366 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39cebc999a51f6204f88ac21e14a162e56815378

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Citations in the corpus

An analysis of electrical impedance tomography with applications to Tikhonov regularization (2012, methodology): “We study the linearized and fully nonlinear models separately, by capitalizing on recent progress on nonsmooth regularization [5, 8, 24, 28, 34, 35, 46, 50].”
Regularization of linear inverse problems with total generalized variation (2014, background): “It is worth mentioning that, as the result of the great interest in the recent years, the study of regularization functionals in general, non-reflexive Banach spaces is now a wellestablished part of the theory of inverse problems (see, for instance, [15, 23, 25] and the references therein).”
Convergence and regularization results for optimal control problems with sparsity functional (2011, methodology): “Convergence rates for non-smooth regularization of inverse problems are proven in Hofmann, Kaltenbacher, Pöschl, Scherzer [10].”
On the second-order asymptotical regularization of linear ill-posed inverse problems (2018, neutral): “Such smoothness assumptions can be expressed by range-type source conditions (cf., e.g., [2]), approximate source conditions (cf. [24]), and variational source conditions occurring in form of variational inequalities (cf. [25]).”
Necessary and sufficient conditions for linear convergence of ℓ1‐regularization (2011, result): “In addition, we compare these results with general convergence rates results with respect to the Bregman distance (see [6, 23, 24, 25, 26, 28, 29, 31, 32, 34]).”
Convergence rates and source conditions for Tikhonov regularization with sparsity constraints (2008, background): “Recently regularization with sparsity constraints has become popular due to the influential paper [9].”
Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces (2014, background): “[19, 25, 26]), and summarize some mathematical consequences of those standard assumptions.”
REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM (2009, background): “Iterative regularization of both linear and nonlinear inverse problems and convergence rates are discussed in Hilbert space setting in [9, 33, 35, 65, 66] and in Banach space setting in [34, 44, 45, 48, 76, 77, 78].”
Regularization with non-convex separable constraints (2009, methodology): “As already noticed in [25] the case q = 1 provides exact penalization, i.”
An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems (2010, background): “In recent publications (see [12, 13, 16, 25]) variational inequalities of the form”
Inverse problems in spaces of measures (2013, background): “While the basic regularization theory of inverse problems is well-established in Hilbert spaces [22] and for linear problems in Banach spaces [34], recent interest focuses on specific linear and non-linear problems usually set in Banach spaces [4,9,21,31,32,38,43,44,49].”
On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems (2010, methodology): “This article is organized as follows: In Section 2 we describe the Tikhonov-type regularization for the stable approximate solution of nonlinear ill-posed operator equations in a Banach space setting under basic assumptions which follow the corresponding assumptions of the papers [6,7].”
Exact Support Recovery for Sparse Spikes Deconvolution (2013, neutral): “Inverse problem regularization over the space of measures is now well understood (see for instance [29, 4]), and requires to perform variational analysis over a non-reﬂexive Banach space (as in [22]), which leads to some mathematical technicalities.”