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Parallel ILU preconditioners in GPU computation

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Abstract

Accelerating large-scale linear solvers is always crucial for scientific research and industrial applications. In this regard, preconditioners play a key role in improving the performance of iterative linear solvers. This paper presents a summary and review of our work about the development of parallel ILU preconditioners on GPUs. The mechanisms of ILU(0), ILU(k), ILUT, enhanced ILUT, and block-wise ILU(k) are reviewed and analyzed, which give a clear guidance in the development of iterative linear solvers. ILU(0) is the most commonly used preconditioner, and the nonzero pattern of its matrix is exactly the same as the original matrix to be solved. ILU(k) uses k levels to control the pattern of its preconditioner matrix. ILUT selects entries for its preconditioner matrix by setting thresholds without considering its original matrix pattern. In addition to point-wise ILU preconditioners, a block-wise ILU(k) preconditioner is designed delicately in support of block-wise matrices. In implementation, the RAS (Restricted Additive Schwarz) method is adopted to optimize the parallel structure of a preconditioner matrix. Coupling with the configuration parameters of ILU preconditioners, a complex situation appears in the parallel solution process, so decoupled algorithms are adopted. These algorithms are implemented and tested on NVIDIA GPUs. The experiment results show that a single-GPU implementation can speed up an ILU preconditioner by a factor of 10, compared to traditional CPU implementation. The results also show that the ILU(0) has better speedup than ILU(k) but slower convergence than ILU(k). Level k of ILU(k) and threshold (p, t) of ILUT are effective adjustment factors for controlling the equilibrium point between acceleration and convergence for ILU(k) and ILUT, respectively. All these ILU preconditioners are characterized and compared in this work, which shows a clear picture and numerical insights for practitioners in the ILU family.

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Acknowledgements

The support of Department of Chemical and Petroleum Engineering, University of Calgary and Reservoir Simulation Group, is gratefully acknowledged. The research is partly supported by NSERC/AIEE/Foundation CMG and AITF Chairs.

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Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Hui Liu, Zhangxin Chen, Bo Yang, Ruijian He & Min Yang

  2. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Wenyuan Liao

  3. College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, China

    Yan Chen & Xuhong Tian

  4. Biomedical Engineering Department, Stony Brook University, Stony Brook, NY, 11794, USA

    Peng Zhang

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  1. Yan Chen
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  2. Xuhong Tian
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Correspondence to Zhangxin Chen or Bo Yang.

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Chen, Y., Tian, X., Liu, H. et al. Parallel ILU preconditioners in GPU computation. Soft Comput 22, 8187–8205 (2018). https://doi.org/10.1007/s00500-017-2764-7

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