Variational regularization theory based on image space approximation rates. (10th May 2021)

• Record Type: Journal Article
• Authors: Miller, Philip
• Is Part Of: Inverse problems. Volume 37:Number 6(2021)
• Publication Date: 2021-05-10
• Subjects: regularization -- convergence rates -- real interpolation -- source conditions -- converse results; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abf5bb
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for 0, 2, q - and 0, p, p -penalties without restrictions on p, q ∈ (1, ∞). Finally we prove equivalence of Hölder-type variational source conditions, bounds on the defect of the Tikhonov functional, and image space approximation rates.