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

TL;DR

This paper introduces a hierarchical block multi-color ordering method that enhances the parallelization and vectorization of sparse triangular solvers within the ICCG method, leading to improved performance.

Contribution

The paper presents a novel parallel ordering technique that combines block multi-color ordering with hierarchical structure to optimize sparse triangular solver performance.

Findings

Outperforms conventional methods in 13 out of 15 test cases

Enables vectorized parallel forward and backward substitutions

Reduces thread synchronization while maintaining fast convergence

Abstract

In this paper, we propose a new parallel ordering method to vectorize and parallelize the sparse triangular solver, which is called hierarchical block multi-color ordering. In this method, the parallel forward and backward substitutions can be vectorized while preserving the advantages of block multi-color ordering, that is, fast convergence and fewer thread synchronizations. To evaluate the proposed method in a parallel ICCG (Incomplete Cholesky Conjugate Gradient) solver, numerical tests were conducted using five test matrices on three types of computational nodes. The numerical results indicate that the proposed method outperforms the conventional block and nodal multi-color ordering methods in 13 out of 15 test cases, which confirms the effectiveness of the method.