RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems

Abstract

We introduce a randomized algorithm, namely 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, rchol samples a subset of the edges in the clique. We prove rchol is breakdown free and apply it to solving linear systems with symmetric diagonallydominant matrices. In addition, we parallelize rchol based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated 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 × 1024 × 1024 grid, or one billion unknowns.