Convergence rates of a dual gradient method for constrained linear ill-posed problems

TL;DR

This paper analyzes the convergence rates of a dual gradient method with a strongly convex penalty for solving constrained linear ill-posed problems, providing theoretical results under source conditions and exploring acceleration and applications.

Contribution

The paper introduces convergence rate analysis for a dual gradient method with a strongly convex penalty in ill-posed problems, including acceleration techniques and application scenarios.

Findings

Convergence rates are established under variational source conditions.

The method's performance is analyzed with both a priori and discrepancy-based stopping rules.

Acceleration of the dual gradient method is also considered.

Abstract

In this paper we consider a dual gradient method for solving linear ill-posed problems $Ax = y$, where $A : X \to Y$ is a bounded linear operator from a Banach space $X$ to a Hilbert space $Y$. A strongly convex penalty function is used in the method to select a solution with desired feature. Under variational source conditions on the sought solution, convergence rates are derived when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. We also consider an acceleration of the method as well as its various applications.