A converse result for Banach space convergence rates in Tikhonov-type\n convex regularization of ill-posed linear equations

2017/12/05 by Jens Flemming, Flemming, Jens · 1 citation
Mathematics · #47A52 #65J22 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1712.01499

openalex publication_date 2017/12/05 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider Tikhonov-type variational regularization of ill-posed linear\noperator equations in Banach spaces with general convex penalty functionals.\nUpper bounds for certain error measures expressing the distance between exact\nand regularized solutions, especially for Bregman distances, can be obtained\nfrom variational source conditions. We prove that such bounds are optimal in\ncase of twisted Bregman distances, that is, the rate function is also an\nasymptotic lower bound for the error measure. This result extends existing\nconverse results from Hilbert space settings to Banach spaces without adhering\nto spectral theory.\n

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