A modified discrepancy principle to attain optimal convergence rates under unknown noise. (11th August 2021)

• Record Type: Journal Article
• Authors: Jahn, Tim
• Is Part Of: Inverse problems. Volume 37:Number 9(2021)
• Publication Date: 2021-08-11
• Subjects: statistical inverse problems -- discrepancy principle -- convergence -- optimality -- spectral cut-off; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac1775
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right-hand side are available. A natural approach is to take the average of the measurements as an approximation of the right-hand side and to estimate the data error as the inverse of the square root of the number of measurements. We calculate the optimal convergence rate (as the number of measurements tends to infinity) under classical source conditions and introduce a modified discrepancy principle, which asymptotically attains this rate.