Consider the linear ill-posed problems of the form $\sum_{i=1}^{b} A_i x_i =y$, where, for each $i$, $A_i$ is a bounded linear operator between two Hilbert spaces $X_i$ and ${\mathcal Y}$. When $b$ is huge, solving the problem by an iterative method using the full gradient at each iteration step is both time-consuming and memory insufficient. Although randomized block coordinate decent (RBCD) method has been shown to be an efficient method for well-posed large-scale optimization problems with a small amount of memory, there still lacks a convergence analysis on the RBCD method for solving ill-posed problems. In this paper, we investigate the convergence property of the RBCD method with noisy data under either {\it a priori} or {\it a posteriori} stopping rules. We prove that the RBCD method combined with an {\it a priori} stopping rule yields a sequence that converges weakly to a solution of the problem almost surely. We also consider the early stopping of the RBCD method and demonstrate that the discrepancy principle can terminate the iteration after finite many steps almost surely. For a class of ill-posed problems with special tensor product form, we obtain strong convergence results on the RBCD method. Furthermore, we consider incorporating the convex regularization terms into the RBCD method to enhance the detection of solution features. To illustrate the theory and the performance of the method, numerical simulations from the imaging modalities in computed tomography and compressive temporal imaging are reported.

翻译：考虑形式为 $\sum_{i=1}^{b} A_i x_i =y$ 的线性不适定问题，其中对于每个 $i$，$A_i$ 是两个希尔伯特空间 $X_i$ 与 ${\mathcal Y}$ 之间的有界线性算子。当 $b$ 极大时，使用每次迭代都计算全梯度的迭代方法求解该问题既耗时又内存不足。尽管随机块坐标下降（RBCD）方法已被证明是求解大规模适定优化问题的一种内存需求小的高效方法，但目前仍缺乏对 RBCD 方法求解不适定问题的收敛性分析。本文研究了在噪声数据下，结合先验或后验停止准则的 RBCD 方法的收敛性质。我们证明了，结合先验停止准则的 RBCD 方法产生的序列几乎必然弱收敛于问题的一个解。我们还考虑了 RBCD 方法的提前停止，并证明了在几乎必然有限步迭代后，残差原理能够终止迭代。对于一类具有特殊张量积形式的不适定问题，我们获得了 RBCD 方法的强收敛结果。此外，我们考虑将凸正则化项引入 RBCD 方法以增强对解特征的识别。为说明理论及方法的性能，文中报告了来自计算机断层扫描和压缩时间成像等成像模态的数值模拟结果。