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：近期，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规模高达$10^{108} \times 10^{108}$的矩阵特征值问题。迄今为止，其成功的完整数学解释仍难以捉摸。此外，这些方法尚未推广至线性方程组求解。本文提出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。我们对该新算法进行了完整的数学分析，证明其收敛速度快于蒙特卡洛方法，并论证了即使解向量本身无法完整存储时该方案的适用性。