Exploiting the memory hierarchy in sequential and parallel sparse Cholesky factorization

Cholesky factorization of large sparse matrices is an extremely important computation, arising in a wide range of domains including linear programming, finite element analysis, and circuit simulation. This thesis investigates crucial issues for obtaining high performance for this computation on sequential and parallel machines with hierarchical memory systems. The thesis begins by providing the first thorough analysis of the interaction between sequential sparse Cholesky factorization methods and memory hierarchies. We look at popular existing methods and find that they produce relatively poor memory hierarchy performance. The methods are extended, using blocking techniques, to reuse data in the fast levels of the memory hierarchy. This increased reuse is shown to provide a three-fold speedup over popular existing approaches (e.g., SPARSPAK) on modern workstations. The thesis then considers the use of blocking techniques in parallel sparse factorization. We first describe parallel methods we have developed that are natural extensions of the sequential approach described above. These methods distribute panels (sets of contiguous columns with nearly identical non-zero structures) among the processors. The thesis shows that for small parallel machines, the resulting methods again produce substantial performance improvements over existing methods. A framework is provided for understanding the performance of these methods, and also for understanding the limitations inherent in them. Using this framework, the thesis shows that panel methods are inappropriate for large-scale parallel machines because they do not expose enough concurrency. The thesis then considers rectangular block methods, where the sparse matrix is split both vertically and horizontally. These methods address the concurrency problems of panel methods, but they also introduce a number of complications. Primary among these are issues of choosing blocks that can be manipulated efficiently and structuring a parallel computation in terms of these blocks. The thesis describes solutions to these problems and presents performance results from an efficient block method implementation. The contributions of this work come both from its theoretical foundation for understanding the factors that limit the scalability of panel- and block-oriented methods on hierarchical memory multiprocessors, and from its investigation of practical issues related to the implementation of efficient parallel factorization methods.