We introduce a randomized algorithm, namely %DIFDELCMD < {\tt %%% 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, %DIFDELCMD < {\tt %%% rchol samples a subset of the edges in the clique  . We prove %DIFDELCMD < {\tt %%% rchol is breakdown free and apply it to solving  linear systems with symmetric diagonally-dominant matrices. In addition, we parallelize %DIFDELCMD < {\tt %%% rchol based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated the robustness and the scalability of %DIFDELCMD < {\tt %%% 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 \textbf{one billion unknowns.
We introduce a randomized algorithm, namely %DIFDELCMD < {\tt %%% 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, %DIFDELCMD < {\tt %%% RCHOL keeps a sparse subset of the edges in the clique using a random sampling developed by Spielman and Kyng . We prove %DIFDELCMD < {\tt %%% RCHOL is breakdown free and apply it to solving large sparse linear systems with symmetric diagonally-dominant matrices. In addition, we parallelize %DIFDELCMD < {\tt %%% RCHOL based on the nested dissection ordering for shared-memory machines. Numerical experiments demonstrated the robustness and the scalability of %DIFDELCMD < {\tt %%% 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 \textbf{one billion unknowns.
R
rchol
RCHOL
D
sparse
R
(exact )
exact
D
rchol
R
samples a
RCHOL keeps a sparse
A
using a random sampling developed by Spielman and Kyng
R
rchol
RCHOL
A
large sparse
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rchol
RCHOL
R
rchol
RCHOL
R
one billion
one billion