Optimal Convergence of the Discrepancy Principle for Polynomially and Exponentially Ill-Posed Operators under White Noise. (25th January 2022)

• Record Type: Journal Article
• Authors: Jahn, Tim
• Is Part Of: Numerical functional analysis and optimization. Volume 43:Number 2(2022)
• Page Start: 145
• Page End: 167
• Publication Date: 2022-01-25
• Subjects: Statistical inverse problems -- non-Bayesian approach -- discrepancy principle -- convergence -- optimality; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2013881
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Schmidt operators (and they require a self-similarity condition for the unknown solution x ̂, additional to a classical source condition) or to polynomially ill-posed operators (excluding exponentially ill-posed problems). In this work, we show optimal convergence for a modified discrepancy principle for both polynomially and exponentially ill-posed operators (without further restrictions) solely under either Hölder-type or logarithmic source conditions. In particular, the method includes only a single simple hyper parameter, which does not need to be adapted to the type of ill-posedness.