Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times 10^{108}$. So far, a complete mathematical explanation for their success has proven elusive. Additionally, the methods have not been extended to linear system solves. In this paper we propose a new scheme based on repeated random sparsification that is capable of solving linear systems in extremely high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector itself is too large to store.

翻译：近年来，一类将经典不动点迭代与近似解向量的重复随机稀疏化相结合的算法已成功应用于矩阵规模高达 $10^{108} \times 10^{108}$ 的特征问题。目前，其成功原因的完整数学解释仍然难以捉摸。此外，这些方法尚未扩展到线性系统求解。在本文中，我们提出了一种基于重复随机稀疏化的新方案，能够求解极高维度的线性系统，并对这一新算法给出了完整的数学分析。我们的分析建立了一种优于蒙特卡洛的收敛速度，并论证了即使在解向量本身过大而无法存储的情况下，该方案的适用性。