This paper describes a method of calculating the Schur complement of a sparse positive definite matrix A. The main idea of this approach is to represent matrix A in the form of an elimination tree using a reordering algorithm like METIS and putting columns/rows for which the Schur complement is needed into the top node of the elimination tree. Any problem with a degenerate part of the initial matrix can be resolved with the help of iterative refinement. The proposed approach is close to the "multifrontal" one which was implemented by Ian Duff and others in 1980s. Schur complement computations described in this paper are available in Intel ® Math Kernel Library (Intel ® MKL). In this paper we present the algorithm for Schur complement computations, experiments that demonstrate a negligible increase in the number of elements in the factored matrix, and comparison with existing alternatives.
The paper describes an efficient direct method to solve an equation Ax = b, where A is a sparse matrix, on the Intel ® Xeon Phi TM coprocessor. The main challenge for such a system is how to engage all available threads (about 240) and how to reduce OpenMP * synchronization overhead, which is very expensive for hundreds of threads. The method consists of decomposing A into a product of lower-triangular, diagonal, and upper triangular matrices followed by solves of the resulting three subsystems. The main idea is based on the hybrid parallel algorithm used in the Intel ® Math Kernel Library Parallel Direct Sparse Solver for Clusters [1]. Our implementation exploits a static scheduling algorithm during the factorization step to reduce OpenMP synchronization overhead. To effectively engage all available threads, a three-level approach of parallelization is used. Furthermore, we demonstrate that our implementation can perform up to 100 times better on factorization step and up to 65 times better in terms of overall performance on the 240 threads of the Intel ® Xeon Phi TM coprocessor.
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