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.
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.
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