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}.