Data-driven stochastic asymptotic regularization for ill-posed inverse problems with a posteriori and heuristic stopping
Abstract
A novel Data-driven STochastic Asymptotic Regularization (\texttt{D-STAR}) method is proposed for solving linear and nonlinear ill-posed inverse problems, which incorporates a data-driven drift and a stochastic diffusion mechanism. Both known and unknown noise level cases are accommodated by the method, thereby making the framework applicable to a broader range of practical inverse problems. For known noise level, a modified discrepancy principle is employed, whereas for unknown noise level, a heuristic stopping rule based solely on noisy data is developed. For each stopping rule, an appropriate diffusion term is defined and the well-posedness of the associated stochastic differential equation is demonstrated. The finite termination of \texttt{D-STAR} method under both stopping rules is established and its convergence to the exact solution in the noise-free setting is proved. For the noisy data, stability and the regularization property in the mean-square sense are further established. In addition, the uncertainty quantification of the proposed method and its comparison with the Stochastic Asymptotic Regularization (\texttt{SAR}) method under both rules are provided. Furthermore, it is shown that local minima arising in the Data-Driven Iteratively Regularized Landweber (\texttt{DDIRLI}) method may be escaped. Finally, detailed numerical experiments, including comparisons with existing \texttt{DDIRLI} and \texttt{SAR} methods, are presented on X-ray computed tomography and nonlinear schlieren tomography for both stopping rules, thereby demonstrating the performance of \texttt{D-STAR}.
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BibTeX
@misc{soni2026datadriven,
title = {{Data-driven stochastic asymptotic regularization for ill-posed inverse problems with a posteriori and heuristic stopping}},
author = {Simran Soni and Harshit Bajpai and Ankik Kumar Giri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38279},
eprint = {2610.38279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38279},
}
RIS
TY - UNPB
TI - Data-driven stochastic asymptotic regularization for ill-posed inverse problems with a posteriori and heuristic stopping
AU - Simran Soni
AU - Harshit Bajpai
AU - Ankik Kumar Giri
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38279
UR - https://arxiv.org/abs/2610.38279
AB - A novel Data-driven STochastic Asymptotic Regularization (\texttt{D-STAR}) method is proposed for solving linear and nonlinear ill-posed inverse problems, which incorporates a data-driven drift and a stochastic diffusion mechanism. Both known and unknown noise level cases are accommodated by the method, thereby making the framework applicable to a broader range of practical inverse problems. For known noise level, a modified discrepancy principle is employed, whereas for unknown noise level, a heuristic stopping rule based solely on noisy data is developed. For each stopping rule, an appropriate diffusion term is defined and the well-posedness of the associated stochastic differential equation is demonstrated. The finite termination of \texttt{D-STAR} method under both stopping rules is established and its convergence to the exact solution in the noise-free setting is proved. For the noisy data, stability and the regularization property in the mean-square sense are further established. In addition, the uncertainty quantification of the proposed method and its comparison with the Stochastic Asymptotic Regularization (\texttt{SAR}) method under both rules are provided. Furthermore, it is shown that local minima arising in the Data-Driven Iteratively Regularized Landweber (\texttt{DDIRLI}) method may be escaped. Finally, detailed numerical experiments, including comparisons with existing \texttt{DDIRLI} and \texttt{SAR} methods, are presented on X-ray computed tomography and nonlinear schlieren tomography for both stopping rules, thereby demonstrating the performance of \texttt{D-STAR}.
ER -