Dual gradient flow for solving linear ill-posed problems in Banach spaces

TL;DR

This paper introduces a dual gradient flow method for solving ill-posed linear problems in Banach spaces, analyzing convergence, stopping rules, and providing numerical validation.

Contribution

It proposes a novel dual gradient flow approach for Banach space problems and establishes convergence and rate results under various stopping rules.

Findings

The dual gradient flow effectively approximates solutions with noisy data.

Proper stopping rules ensure convergence to reasonable solutions.

Numerical experiments validate the method's performance.

Abstract

We consider determining the $\R$-minimizing solution of ill-posed problem $A x = y$ for a bounded linear operator $A: X \to Y$ from a Banach space $X$ to a Hilbert space $Y$, where $\R: X \to (-\infty, \infty]$ is a strongly convex function. A dual gradient flow is proposed to approximate the sought solution by using noisy data. Due to the ill-posedness of the underlying problem, the flow demonstrates the semi-convergence phenomenon and a stopping time should be chosen carefully to find reasonable approximate solutions. We consider the choice of a proper stopping time by various rules such as the {\it a priori} rules, the discrepancy principle, and the heuristic discrepancy principle and establish the respective convergence results. Furthermore, convergence rates are derived under the variational source conditions on the sought solution. Numerical results are reported to test the performance of the dual gradient flow.