In this paper, we consider the iterative solution of sparse systems of linear algebraic equations under the condition that sparse matrix-vector products with the coefficient matrix are computed only partially. At the same time, non-computed entries are set to zeros. We assume that both the number of computed entries and their associated row index set are random variables, with the row index set sampled uniformly given the number of computed entries. This model of computations is prevalent to that realized in hybrid cloud computing architectures following the controller-worker distributed model under the influence of straggling workers. We propose a randomized Richardson iterative scheme and a randomized Chebyshev semi-iterative method within this model and prove the sufficient conditions for their convergence in expectation. Numerical experiments verify the presented theoretical results as well as the effectiveness of the proposed schemes on a few sparse matrix problems.

翻译：本文研究稀疏线性代数方程组的迭代求解问题，其计算条件为：系数矩阵的稀疏矩阵-向量乘积仅被部分计算。同时，未计算项被设为零。我们假设计算项的数量及其对应的行索引集均为随机变量，其中行索引集在给定计算项数量的条件下均匀采样。该计算模型广泛存在于采用控制器-工作者分布式模型的混合云计算架构中，且受到延迟工作节点的影响。在此模型下，我们提出一种随机化Richardson迭代格式和一种随机化Chebyshev半迭代方法，并证明了它们在期望意义下收敛的充分条件。数值实验验证了所提出的理论结果，并在若干稀疏矩阵问题上证明了所提方法的有效性。