RCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems

We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given 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 keeps 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. 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 $1024\times 1024 \times 1024$ grid, or one billion unknowns.