A modified discrepancy principle to attain optimal convergence rates under unknown noise

TL;DR

This paper introduces a modified discrepancy principle for linear ill-posed problems with multiple measurements, achieving optimal convergence rates as the number of measurements increases, under classical source conditions.

Contribution

The paper proposes a new modified discrepancy principle that asymptotically attains the optimal convergence rate in ill-posed problems with unknown noise levels.

Findings

The modified discrepancy principle achieves the optimal convergence rate.

Averages of multiple measurements effectively estimate the data error.

The approach is validated under classical source conditions.

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.