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<p>Inverse problems with noisy operators arise in many applications, especially where some key terms to model the forward problem are contaminated by random noise, or the forward operator is approximated by a numerical discretization of the infinite-dimensional problem into a finite-dimensional one or some so-called data-driven methods. In this paper, we develop a Landweber iteration with non-smooth constraints for solving such ill-posed problems. Since the noise levels of the noisy data and operator are usually unavailable or unreliable in practice, we also propose a heuristic parameter choice rule, i.e., the Hanke-Raus rule, to terminate the iteration instead of the discrepancy principle. By imposing appropriate conditions on the noisy operator, we establish the convergence of the method under both stopping rules. Finally, numerical examples including parameter identification of partial differential equation problems and CT image reconstruction are presented to illustrate the effectiveness of the proposed method, especially under the Hanke-Raus rule.</p>

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A Landweber-type iteration with non-smooth convex constraints for ill-posed problems with noisy operators

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Ran Gu,
  • Chaofeng Dong,
  • Wei Wang

摘要

Inverse problems with noisy operators arise in many applications, especially where some key terms to model the forward problem are contaminated by random noise, or the forward operator is approximated by a numerical discretization of the infinite-dimensional problem into a finite-dimensional one or some so-called data-driven methods. In this paper, we develop a Landweber iteration with non-smooth constraints for solving such ill-posed problems. Since the noise levels of the noisy data and operator are usually unavailable or unreliable in practice, we also propose a heuristic parameter choice rule, i.e., the Hanke-Raus rule, to terminate the iteration instead of the discrepancy principle. By imposing appropriate conditions on the noisy operator, we establish the convergence of the method under both stopping rules. Finally, numerical examples including parameter identification of partial differential equation problems and CT image reconstruction are presented to illustrate the effectiveness of the proposed method, especially under the Hanke-Raus rule.