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

TL;DR

This paper proves optimal convergence of a modified discrepancy principle for both polynomially and exponentially ill-posed operators in Hilbert spaces, under simple source conditions and with minimal parameter tuning.

Contribution

It introduces a unified approach that achieves optimal convergence for a broad class of ill-posed problems without restrictive assumptions or complex parameter adjustments.

Findings

Optimal convergence for polynomially and exponentially ill-posed operators

Method requires only a single hyperparameter

Works under simple source conditions

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 $\hat{x}$, 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 H\"older-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.