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<p>In recent years, stochastic algorithms have been introduced to solve ill-posed inverse problems. These algorithms select a random subset of equations during each iteration, displaying excellent scalability and competitive performance in large-scale inverse problems. However, given the inherent ill-posed nature of the underlying problems and the presence of noise in measurement data, these algorithms often exhibit prominent oscillations and display a semi-convergence phenomenon, like all iterative regularization methods. This aspect poses challenges in obtaining an output with good approximation property. In this paper, by leveraging the spirit of the discrepancy principle we propose an a posteriori stopping rule for the stochastic mirror descent method for solving ill-posed inverse problems in Banach spaces. We show that the proposed stopping rule <i>always</i> terminates the method within a finite number of steps and the corresponding outcome converges toward the sought solution <i>almost surely</i> as the noise level approaches zero. Numerical simulations are reported to demonstrate the promising performance.</p>

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On early stopping of stochastic mirror descent method for ill-posed inverse problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月6日

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  • Jing Huang,
  • Qinian Jin,
  • Xiliang Lu,
  • Liuying Zhang

摘要

In recent years, stochastic algorithms have been introduced to solve ill-posed inverse problems. These algorithms select a random subset of equations during each iteration, displaying excellent scalability and competitive performance in large-scale inverse problems. However, given the inherent ill-posed nature of the underlying problems and the presence of noise in measurement data, these algorithms often exhibit prominent oscillations and display a semi-convergence phenomenon, like all iterative regularization methods. This aspect poses challenges in obtaining an output with good approximation property. In this paper, by leveraging the spirit of the discrepancy principle we propose an a posteriori stopping rule for the stochastic mirror descent method for solving ill-posed inverse problems in Banach spaces. We show that the proposed stopping rule always terminates the method within a finite number of steps and the corresponding outcome converges toward the sought solution almost surely as the noise level approaches zero. Numerical simulations are reported to demonstrate the promising performance.