We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given Laplacian matrix (a.k.a., graph Laplacian). From a graph perspective, the exact Cholesky factorization introduces a clique in the underlying graph after eliminating a row/column. By randomization, RCHOL only retains a sparse subset of the edges in the clique using a random sampling developed by Spielman and Kyng. We prove RCHOL is breakdown-free and apply it to solving large sparse linear systems with symmetric diagonally dominant matrices. In addition, we parallelize RCHOL based on the nested dissection ordering for shared-memory machines. We report numerical experiments that demonstrate the robustness and the scalability of 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, a problem that has one billion unknowns.
翻译:我们引入了随机算法, 即 RCHOL, 用于为给定的 Laplacian 矩阵( a.k.a., 图形 Laplacecian ) 构建一个近似Challosky 的系数化。 从图形角度看, 精确的 Challosky 系数化在消除行/ 栏后在底图中引入了球状。 通过随机化, RCHOL 只使用Spielman 和 Kyng 开发的随机抽样, 保留了微小的区际边缘子组。 我们证明 RCHOL 是无损的, 并应用它来用对称对称主矩阵解决大片稀薄线性系统。 此外, 我们根据对共享模拟机器的嵌入解剖令将 RCHOL 平行化 。 我们报告的数字实验显示了 RCHOL 的稳健性和可缩缩放性 。 例如, 我们的平行代码在解决 3D Poisson 方程式的单个节点上, 升至64 线条线条线条, 与 1024\ time 1024\ 1024\ y time 1024 1024 24 y time 1024 1024 1024 兆格上 10 10 10 10 10亿 个 未知 个 问题 未知 未知 。