RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems
Chao Chen, Tianyu Liang, George Biros · SIAM Journal on Scientific Computing · 2021
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 [ private communication, 2020]. 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 three-dimensional Poisson equation, discretized with the 7-point stencil on a 1024 x 1024 x 1024 grid, a problem that has one billion unknowns.