  Direct factorization methods for the solution of large, sparse linear systems
that arise from PDE discretizations are robust, but typically show poor time
and memory scalability for large systems. In this paper, we describe an
efficient sparse, rank-structured Cholesky algorithm for solution of the
positive definite linear system $A x = b$ when $A$ comes from a discretized
partial-differential equation. Our approach combines the efficient memory
access patterns of conventional supernodal Cholesky algorithms with the memory
efficiency of rank-structured direct solvers. For several test problems arising
from PDE discretizations, our method takes less memory than standard sparse
Cholesky solvers and less wall-clock time than standard preconditioned
iterations.
