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The authors explore the use of a sub-block decomposition strategy for parallel sparse Cholesky factorization, in which the sparse matrix is decomposed into rectangular blocks. Such a strategy has enormous theoretical scalability advantages over more traditional column-oriented and panel-oriented decompositions. However, little progress has been made in producing a practical sub-block method. The authors propose and evaluate an approach that is simple to implement, provides slightly higher performance than column (and panel) methods on small parallel machines, and has the potential to provide much higher performance on large parallel machines.