A discretized Predictor–Corrector iterative Tikhonov regularization method via heuristic parameter choice rule

In this paper, we propose a discretized Predictor–Corrector iterative Tikhonov regularization (DPC-ITR) method, which integrates multiscale Galerkin projection for discretization, iterative Tikhonov regularization for inversion, and the Modified Euler Method for time stepping. The proposed DPC-ITR method significantly accelerates computation for ill-posed inverse problems compared to standard iterative Tikhonov methods, while preserving the same order of numerical accuracy. Under specific regularity conditions, we rigorously derive a priori error estimates for the approximate solutions generated by the DPC-ITR method. Furthermore, we propose a novel heuristic parameter choice rule for the DPC-ITR method when applied to linear ill-posed integral equations. The proposed parameter choice rule, under certain conditions, enables the DPC-ITR method to generate approximate solutions that converge at order-optimal rates, as rigorously proven in our analysis. Our numerical experiments confirm that the DPC-ITR method equipped with the proposed heuristic parameter choice rule achieves the theoretically predicted convergence rates while demonstrating improved computational efficiency compared to conventional regularization approaches.