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Direct factorization methods for the solution of large, sparse linear systemsthat arise from PDE discretizations are robust, but typically show poor timeand memory scalability for large systems. In this paper, we describe anefficient sparse, rank-structured Cholesky algorithm for solution of thepositive definite linear system $A x = b$ when $A$ comes from a discretizedpartial-differential equation. Our approach combines the efficient memoryaccess patterns of conventional supernodal Cholesky algorithms with the memoryefficiency of rank-structured direct solvers. For several test problems arisingfrom PDE discretizations, our method takes less memory than standard sparseCholesky solvers and less wall-clock time than standard preconditionediterations.

An Efficient Solver for Sparse Linear Systems Based on Rank-Structured Cholesky Factorization