后验停止规则下大规模不适定问题的随机Krylov投影迭代Tikhonov正则化
Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule
AI总结：
针对大规模线性不适定逆问题，提出RIGKT和RIAT随机迭代正则化框架，结合随机迭代Tikhonov正则化与Krylov子空间投影技术，纳入多种策略及后验停止规则，经分析和实验验证其能可靠重建结构特征。
AI中文摘要：
我们引入了两种新颖的随机迭代正则化框架，即RIGKT和RIAT，用于解决由方程组控制的大规模线性不适定逆问题。所提出的方法将随机迭代Tikhonov正则化与Krylov子空间投影技术相结合，对于一般矩形系统使用Golub-Kahan双对角化（RIGKT），对于方阵使用Arnoldi分解（RIAT）。与现有依赖固定迭代次数的确定性方案不同，我们的框架纳入了随机方程选择、自适应步长策略以及专门针对随机设置的基于全局差异的后验早期停止规则。我们进行了全面的正则化分析，建立了Bregman距离单调性、有限终止、精确数据收敛和噪声下的路径稳定性。此外，我们证明了停止迭代几乎必然且在均方意义下收敛到真实解，确立了严格的正则化性质。数值实验表明，RIGKT和RIAT在各种噪声情况下都能可靠地重建结构特征。
英文摘要：
We introduce two novel randomized iterative regularization frameworks, termed \texttt{RIGKT} and \texttt{RIAT}, for solving large-scale linear ill-posed inverse problems governed by systems of equations. The proposed methods combine randomized iterated Tikhonov regularization with Krylov subspace projection techniques, utilizing Golub--Kahan bidiagonalization for general rectangular systems (\texttt{RIGKT}) and Arnoldi decomposition for square systems (\texttt{RIAT}). Unlike existing deterministic schemes that rely on fixed iteration counts, our framework incorporates randomized equation selection, an adaptive step-size strategy, and a global, discrepancy-based a posteriori early-stopping rule tailored specifically to the stochastic setting. We present a comprehensive regularization analysis establishing Bregman-distance monotonicity, finite termination, exact-data convergence, and pathwise stability under noise. Furthermore, we prove that the stopped iterates converge almost surely and in the mean-square sense to the true solution, establishing a rigorous regularization property. To the best of our knowledge, this is the first theoretical framework to simultaneously account for randomization, Krylov-subspace dimension reduction, and implementable early stopping. Numerical experiments involving two-dimensional X-ray computed tomography (CT) and image deblurring demonstrate that \texttt{RIGKT} and \texttt{RIAT} reliably reconstruct structural features across various noise regimes.
