Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise

2021/04/13 by Tim Jahn, Jahn, Tim · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Non-Destructive Testing Techniques #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Ultrasonics and Acoustic Wave Propagation

paper · pdf · doi:10.48550/arxiv.2104.06184

openalex publication_date 2021/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Convergence of generalized cross-validation with applications to ill-posed integral equations

Related

• A discrepancy principle for equations with monotone continuous operators
• Analysis of the discrepancy principle for Tikhonov regularization under low order source conditions
• On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems
• Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces
• A new regularization method for linear exponentially ill-posed problems