Dual gradient method for ill-posed problems using multiple repeated measurement data

TL;DR

This paper introduces a dual gradient method for solving linear ill-posed problems using multiple repeated measurements, providing convergence analysis and numerical validation for the approach.

Contribution

It proposes a novel dual gradient method leveraging multiple data sets for ill-posed problems, with convergence guarantees and practical stopping rules.

Findings

Convergence rates established under variational source conditions.

Method effectively reconstructs solutions from repeated measurements.

Numerical experiments demonstrate the method's performance.

Abstract

We consider determining $\R$-minimizing solutions of linear ill-posed problems $A x = y$, where $A: {\mathscr X} \to {\mathscr Y}$ is a bounded linear operator from a Banach space ${\mathscr X}$ to a Hilbert space ${\mathscr Y}$ and ${\mathcal R}: {\mathscr X} \to [0, \infty]$ is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of $y$ are available, we consider a dual gradient method to reconstruct the ${\mathcal R}$-minimizing solution using the average of these data. By terminating the method by either an {\it a priori} stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method.