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

Qinian Jin, Wei Wang

TL;DR

This work tackles the ill-posed linear problem $A x = y$ in Banach spaces by seeking the $\mathcal{R}$-minimizing solution, where $\mathcal{R}$ is strongly convex. It introduces a dual gradient flow in the Banach setting, yielding a continuous regularization method with semi-convergence properties and a computational primitive $x(t)=\nabla\mathcal{R}^*(A^*\lambda(t))$ driven by noisy data $y^\delta$, $\frac{d}{dt}\lambda(t)= y^\delta - A x(t)$. The authors establish well-posedness, monotonic residual behavior, and a suite of stopping rules—an a priori rule, the discrepancy principle, and a heuristic discrepancy principle—proving convergence and convergence rates under variational source conditions, and they derive duality-based estimates that connect primal and dual progress. Numerical experiments on a Fredholm integral equation and 2D tomography demonstrate the method’s practical performance, showing robust reconstructions and clear semi-convergence behavior under both discrepancy-based and heuristic stopping rules. The results highlight the applicability of convex-analysis tools in Banach spaces for designing stable, parameter-choice strategies in inverse problems.

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.

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

TL;DR

This work tackles the ill-posed linear problem in Banach spaces by seeking the -minimizing solution, where is strongly convex. It introduces a dual gradient flow in the Banach setting, yielding a continuous regularization method with semi-convergence properties and a computational primitive driven by noisy data , . The authors establish well-posedness, monotonic residual behavior, and a suite of stopping rules—an a priori rule, the discrepancy principle, and a heuristic discrepancy principle—proving convergence and convergence rates under variational source conditions, and they derive duality-based estimates that connect primal and dual progress. Numerical experiments on a Fredholm integral equation and 2D tomography demonstrate the method’s practical performance, showing robust reconstructions and clear semi-convergence behavior under both discrepancy-based and heuristic stopping rules. The results highlight the applicability of convex-analysis tools in Banach spaces for designing stable, parameter-choice strategies in inverse problems.

Abstract

We consider determining the -minimizing solution of ill-posed problem for a bounded linear operator from a Banach space to a Hilbert space , where 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.
Paper Structure (7 sections, 13 theorems, 139 equations, 2 figures, 2 tables)

This paper contains 7 sections, 13 theorems, 139 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Convergence analysis
  3. Key properties of the dual gradient flow
  4. A priori parameter choice
  5. The discrepancy principle
  6. The heuristic discrepancy principle
  7. Numerical results

Key Result

Proposition 2.1

Let $X$ be a Banach space and let $\mathcal{R}: X \to (-\infty, \infty]$ be a proper, lower semi-continuous, strongly convex function as stated in Assumption dgm.ass1 (ii). Then $\emph{dom}(\mathcal{R}^*) = X^*$, $\mathcal{R}^*$ is Fréchet differentiable and its gradient $\nabla \mathcal{R}^*$ maps for all $\xi', \xi \in X^*$.

Figures (2)

  • Figure 1: The reconstructed results for Example \ref{['ex1']} using noisy data with noise level $\delta = 0.01$.
  • Figure 2: The reconstructed results for Example \ref{['ex2']} using noisy data with relative noise level $\delta_{rel} = 0.01$.

Theorems & Definitions (27)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • ...and 17 more