Convergence rates for ℓ1-regularization without injectivity-type assumptions. (5th July 2016)

• Record Type: Journal Article
• Authors: Flemming, Jens
• Is Part Of: Inverse problems. Volume 32:Number 9(2016:Sep.)
• Publication Date: 2016-07-05
• Subjects: sparsity promoting regularization -- Tikhonov regularization -- non-injective -- variational source condition -- convergence rates; 65J20 -- 47A52; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/9/095001
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: Existing convergence rate results for sparsity promoting regularization of Tikhonov-type rely on injectivity of the considered operator or at least on slightly weakened injectivity assumptions (finite basis injectivity or restricted isometry property). We extend such results to non-injective operators by formulating a suitable variational source condition, which then is characterized in the language of range conditions with respect to the range of the adjoint operator. As a special case we consider operator equations with uniquely determined 1-norm minimizing solutions. Based on the developed characterization of a variational source condition we also provide convergence rates for the case that solutions are not sparse.