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A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems

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Abstract

Solving block-tridiagonal systems is one of the key issues in numerical simulations of many scientific and engineering problems. Non-zero elements are mainly concentrated in the blocks on the main diagonal for most block-tridiagonal matrices, and the blocks above and below the main diagonal have little non-zero elements. Therefore, we present a solving method which mixes direct and iterative methods. In our method, the submatrices on the main diagonal are solved by the direct methods in the iteration processes. Because the approximate solutions obtained by the direct methods are closer to the exact solutions, the convergence speed of solving the block-tridiagonal system of linear equations can be improved. Some direct methods have good performance in solving small-scale equations, and the sub-equations can be solved in parallel. We present an improved algorithm to solve the sub-equations by thread blocks on GPU, and the intermediate data are stored in shared memory, so as to significantly reduce the latency of memory access. Furthermore, we analyze cloud resources scheduling model and obtain ten block-tridiagonal matrices which are produced by the simulation of the cloud-computing system. The computing performance of solving these block-tridiagonal systems of linear equations can be improved using our method.

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Acknowledgments

The authors deeply appreciate the anonymous reviewers for their comments on the manuscript. The research was partially funded by the Key Program of National Natural Science Foundation of China (Grant Nos. 61133005 and 61432005), the National Natural Science Foundation of China (Grant Nos. 61370095, 61472124, and 61572175), and the Science and technology project of Hunan Province (Grant No. 2015SK20062).

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Authors and Affiliations

  1. College of Information Science and Engineering, Hunan University, Changsha, 410008, Hunan, China

    Wangdong Yang, Kenli Li & Keqin Li

  2. Department of Computer Science, State University of New York, New Paltz, NY, 12561, USA

    Keqin Li

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  1. Wangdong Yang
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Correspondence to Wangdong Yang.

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Yang, W., Li, K. & Li, K. A parallel solving method for block-tridiagonal equations on CPU–GPU heterogeneous computing systems. J Supercomput 73, 1760–1781 (2017). https://doi.org/10.1007/s11227-016-1881-x

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