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Robustness and Accuracy in Pipelined Bi-Conjugate Gradient Stabilized Method: A Comparative Study

Mykhailo Havdiak, Jose I. Aliaga, Roman Iakymchuk

TL;DR

The paper addresses robustness and accuracy of the pipelined BiCGStab for unsymmetric linear systems on distributed memory architectures. It proposes an ExBLAS-based reproducible framework to replace the residual replacement technique and compares BiCGStab, p-BiCGStab, p-BiCGStabExBLAS, and p-BiCGStabRR on matrices from the SuiteSparse Matrix Collection. Results indicate that the ExBLAS approach provides reproducible, reliable convergence and often better numerical stability than the baseline pipelined method, while BiCGStab generally remains the most stable; residual replacement remains situationally advantageous but can be slower and more complex. The findings support ExBLAS as a viable path to improve reliability and scalability of parallel Krylov solvers for challenging non-symmetric problems in HPC contexts.

Abstract

In this article, we propose an accuracy-assuring technique for finding a solution for unsymmetric linear systems. Such problems are related to different areas such as image processing, computer vision, and computational fluid dynamics. Parallel implementation of Krylov subspace methods speeds up finding approximate solutions for linear systems. In this context, the refined approach in pipelined BiCGStab enhances scalability on distributed memory machines, yielding to substantial speed improvements compared to the standard BiCGStab method. However, it's worth noting that the pipelined BiCGStab algorithm sacrifices some accuracy, which is stabilized with the residual replacement technique. This paper aims to address this issue by employing the ExBLAS-based reproducible approach. We validate the idea on a set of matrices from the SuiteSparse Matrix Collection.

Robustness and Accuracy in Pipelined Bi-Conjugate Gradient Stabilized Method: A Comparative Study

TL;DR

The paper addresses robustness and accuracy of the pipelined BiCGStab for unsymmetric linear systems on distributed memory architectures. It proposes an ExBLAS-based reproducible framework to replace the residual replacement technique and compares BiCGStab, p-BiCGStab, p-BiCGStabExBLAS, and p-BiCGStabRR on matrices from the SuiteSparse Matrix Collection. Results indicate that the ExBLAS approach provides reproducible, reliable convergence and often better numerical stability than the baseline pipelined method, while BiCGStab generally remains the most stable; residual replacement remains situationally advantageous but can be slower and more complex. The findings support ExBLAS as a viable path to improve reliability and scalability of parallel Krylov solvers for challenging non-symmetric problems in HPC contexts.

Abstract

In this article, we propose an accuracy-assuring technique for finding a solution for unsymmetric linear systems. Such problems are related to different areas such as image processing, computer vision, and computational fluid dynamics. Parallel implementation of Krylov subspace methods speeds up finding approximate solutions for linear systems. In this context, the refined approach in pipelined BiCGStab enhances scalability on distributed memory machines, yielding to substantial speed improvements compared to the standard BiCGStab method. However, it's worth noting that the pipelined BiCGStab algorithm sacrifices some accuracy, which is stabilized with the residual replacement technique. This paper aims to address this issue by employing the ExBLAS-based reproducible approach. We validate the idea on a set of matrices from the SuiteSparse Matrix Collection.
Paper Structure (4 sections, 3 figures, 4 tables, 2 algorithms)

This paper contains 4 sections, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Reproducibility of BiCGStab and pipelined BiCGStab
  3. Experimental Results
  4. Conclusion

Figures (3)

  • Figure 1: Number of iterations required by various BiCGStab-like methods to achieve a specified tolerance ($10^{-6}, 10^{-9}, 10^{-13}$). p-BiCGStabRR stands for the pipelined version of the BiCGStab method with residual replacement; P-BiCGStabExBLAS refers to the method with ExBLAS.
  • Figure 2: Residual history of the four BiCGStab-like methods: BiCGStab, pipelined BiCGStab (p-BiCGStab), pipelined BiCGStab with ExBLAS (p-BiCGStabExBLAS), pipelined BiCGStab with residual replacement (p-BiCGStabRR); $tol = 10^{-13}$.
  • Figure 3: Runtime comparison of BiCGStab-like methods on two SuiteSparse matrices, namely Queen_4147 with $316,548,962$ nnz and s3dkq4m2 with $4,427,725$ nnz, using a various number of MPI processes; $tol=10^{-6}$ for Queen_4147 and $tol=10^{-9}$ for s3dkq4m2.