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Convergence rates in ℓ 1 regularization if the sparsity assumption fails
 Inverse Problems
"... Variational sparsity regularization based on ℓ 1norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization paramet ..."
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Variational sparsity regularization based on ℓ 1norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of illposed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter.
ON THE INTERPLAY OF BASIS SMOOTHNESS AND SPECIFIC RANGE CONDITIONS OCCURRING IN SPARSITY REGULARIZATION
, 2013
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The impact of a curious type of smoothness conditions on convergence rates in ℓ1regularization
 Eurasian Journal of Mathematical and Computer Applications
"... Abstract Tikhonovtype regularization of linear and nonlinear illposed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the 1 norm is used as penalty term, the 1 regulariz ..."
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Abstract Tikhonovtype regularization of linear and nonlinear illposed problems in abstract spaces under sparsity constraints gained relevant attention in the past years. Since under some weak assumptions all regularized solutions are sparse if the 1 norm is used as penalty term, the 1 regularization was studied by numerous authors although the nonreflexivity of the Banach space 1 and the fact that such penalty functional is not strictly convex lead to serious difficulties. We consider the case that the sparsity assumption is narrowly missed. This means that the solutions may have an infinite number of nonzero but fast decaying components. For that case we formulate and prove convergence rates results for the 1 regularization of nonlinear operator equations. In this context, we outline the situations of Hölder rates and of an exponential decay of the solution components. MSC2010 subject classification: 47J06, 65J20, 47A52, 49J40
On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A DiscrepancyBased Strategy
, 2014
"... We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regulari ..."
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We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancybased choice for these quantities by applying a relaxed version of Morozov’s discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers.