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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 $1024times 1024 times 1024$ grid, a problem that has one billion unknowns.
We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity.
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits th
The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frame
Some new rigorous perturbation bounds for the generalized Cholesky factorization with normwise or componentwise perturbations in the given matrix are obtained, where the componentwise perturbation has the form of backward rounding error for the gener
We propose to compute a sparse approximate inverse Cholesky factor $L$ of a dense covariance matrix $Theta$ by minimizing the Kullback-Leibler divergence between the Gaussian distributions $mathcal{N}(0, Theta)$ and $mathcal{N}(0, L^{-top} L^{-1})$,