We present a data-assisted iterative regularization method for solving ill-posed inverse problems. The proposed approach, termed \texttt{IRMGL+\(\Psi\)}, integrates classical iterative techniques with a data-driven regularization term realized through an iteratively updated graph Laplacian. Our method commences by computing a preliminary solution using any suitable reconstruction method, which then serves as the basis for constructing the initial graph Laplacian. The solution is subsequently refined through an iterative process, where the graph Laplacian is simultaneously recalibrated at each step to effectively capture the evolving structure of the solution. A key innovation of this work lies in the formulation of this iterative scheme and the rigorous justification of the classical discrepancy principle as a reliable early stopping criterion specifically tailored to the proposed method. Under standard assumptions, we establish stability and convergence results for the scheme when the discrepancy principle is applied. Furthermore, we demonstrate the robustness and effectiveness of our method through numerical experiments utilizing four distinct initial reconstructors $\Psi$: the adjoint operator (Adj), filtered back projection (FBP), total variation (TV) denoising, and standard Tikhonov regularization (Tik). It is observed that \texttt{IRMGL+Adj} demonstrates a distinct advantage over the other initializers, producing a robust and stable approximate solution directly from a basic initial reconstruction.

翻译：本文提出了一种数据辅助的迭代正则化方法，用于求解不适定反问题。所提出的方法称为 \texttt{IRMGL+\(\Psi\)}，它将经典迭代技术与数据驱动的正则化项相结合，该正则化项通过迭代更新的图拉普拉斯算子实现。我们的方法首先使用任何合适的重建方法计算一个初步解，该解随后作为构建初始图拉普拉斯算子的基础。接着通过一个迭代过程对解进行细化，在此过程中，图拉普拉斯算子在每一步同时被重新校准，以有效捕捉解的演化结构。本工作的一个关键创新在于该迭代方案的构建，以及对经典偏差原理作为专门为本方法定制的可靠早停准则的严格论证。在标准假设下，我们建立了应用偏差原理时该方案的稳定性和收敛性结果。此外，我们通过使用四种不同初始重建器 $\Psi$ 的数值实验，证明了我们方法的鲁棒性和有效性：伴随算子（Adj）、滤波反投影（FBP）、全变分（TV）去噪和标准吉洪诺夫正则化（Tik）。实验观察到，\texttt{IRMGL+Adj} 相较于其他初始器展现出明显优势，能够直接从基础的初始重建中产生鲁棒且稳定的近似解。