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The paper describes an efficient direct method to solve an equation Ax = b, where A is a sparse matrix, on the Intel® Xeon PhiTM 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 PhiTM coprocessor.

applied mathematics

Intel&lt;sup&gt;&amp;reg;&lt;/sup&gt; Math Kernel Library PARDISO&lt;sup&gt;*&lt;/sup&gt; for Intel&lt;sup&gt;&amp;reg;&lt;/sup&gt; Xeon Phi&lt;sup&gt;TM&lt;/sup&gt; Manycore Coprocessor