We introduce a randomized algorithm, namely {\tt rchol}, to construct an approximate Cholesky factorization for a given sparse Laplacian matrix (a.k.a., graph Laplacian). The (exact) Cholesky factorization for the matrix introduces a clique in the associated graph after eliminating every row/column. By randomization, {\tt rchol} samples a subset of the edges in the clique. We prove {\tt rchol} is breakdown free and apply it to solving linear systems with symmetric diagonally-dominant matrices. In addition, we parallelize {\tt rchol} based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated the robustness and the scalability of {\tt rchol}. For example, our parallel code scaled up to 64 threads on a single node for solving the 3D Poisson equation, discretized with the 7-point stencil on a $1024\times 1024 \times 1024$ grid, or \textbf{one billion} unknowns.

中文翻译：

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RCHOL：用于求解 SDD 线性系统的随机 Cholesky 分解

我们引入了一种随机算法，即 {\tt rchol}，为给定的稀疏拉普拉斯矩阵（又名拉普拉斯图）构造近似的 Cholesky 分解。矩阵的（精确）Cholesky 分解在消除每一行/列后在相关图中引入了一个集团。通过随机化，{\tt rchol} 对集团中的边的一个子集进行采样。我们证明 {\tt rchol} 是无故障的，并将其应用于求解具有对称对角主导矩阵的线性系统。此外，我们基于共享内存机器的嵌套解剖顺序并行化 {\tt rchol}。数值实验证明了 {\tt rchol} 的鲁棒性和可扩展性。例如，我们的并行代码在单个节点上扩展到 64 个线程来求解 3D 泊松方程，