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In this paper, we propose a new parallel ordering method to vectorize andparallelize the sparse triangular solver, which is called hierarchical blockmulti-color ordering. In this method, the parallel forward and backwardsubstitutions can be vectorized while preserving the advantages of blockmulti-color ordering, that is, fast convergence and fewer threadsynchronizations. To evaluate the proposed method in a parallel ICCG(Incomplete Cholesky Conjugate Gradient) solver, numerical tests were conductedusing five test matrices on three types of computational nodes. The numericalresults indicate that the proposed method outperforms the conventional blockand nodal multi-color ordering methods in 13 out of 15 test cases, whichconfirms the effectiveness of the method.

Hierarchical Block Multi-Color Ordering: A New Parallel Ordering Method  for Vectorization and Parallelization of the Sparse Triangular Solver in the  ICCG Method