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Abstract <p>In solving various problems of mathematical physics, it is often necessary to repeatedly find the solution of systems of linear algebraic equations. To solve these systems, various iterative numerical methods can be employed, such as Krylov subspace and/or multigrid methods. One po- pular approach to reduce computation time is the utilization of graphics processing units (GPUs). In this work, we investigate the efficiency of using GPUs to accelerate the solution of systems of linear algebraic equations employing the Bi-Conjugate Gradient Stabilized method with an algebraic multigrid preconditioner and various smoothers. These methods were implemented within the XAMG library for both GPU and CPU architectures. Testing was performed on both a local workstation and a cluster node, including efficiency comparisons between the sparse matrix-vector multiplication implementations in the XAMG library and those in the cuSPARSE library. The testing performed shows that the average speedup of the SLAEs solution depends on the method used and varies from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(7\)</EquationSource> <!--LobJMat2561526Kuprii-m1--> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(9\times\)</EquationSource> <!--LobJMat2561526Kuprii-m2--> </InlineEquation> on the workstation and from <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(5.5\times\)</EquationSource> <!--LobJMat2561526Kuprii-m3--> </InlineEquation> to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(7\times\)</EquationSource> <!--LobJMat2561526Kuprii-m4--> </InlineEquation> on the cluster node. It is also demonstrated that solving SLAEs on GPU in the XAMG library is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(1.25\times\)</EquationSource> <!--LobJMat2561526Kuprii-m5--> </InlineEquation> faster than in the <i>hypre</i> library.</p>

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Evaluating Efficiency of Using Graphics Processing Units for Solving Systems of Linear Algebraic Equations

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • R. M. Kuprii,
  • B. I. Krasnopolsky,
  • K. A. Zhukov

摘要

Abstract

In solving various problems of mathematical physics, it is often necessary to repeatedly find the solution of systems of linear algebraic equations. To solve these systems, various iterative numerical methods can be employed, such as Krylov subspace and/or multigrid methods. One po- pular approach to reduce computation time is the utilization of graphics processing units (GPUs). In this work, we investigate the efficiency of using GPUs to accelerate the solution of systems of linear algebraic equations employing the Bi-Conjugate Gradient Stabilized method with an algebraic multigrid preconditioner and various smoothers. These methods were implemented within the XAMG library for both GPU and CPU architectures. Testing was performed on both a local workstation and a cluster node, including efficiency comparisons between the sparse matrix-vector multiplication implementations in the XAMG library and those in the cuSPARSE library. The testing performed shows that the average speedup of the SLAEs solution depends on the method used and varies from \(7\) to \(9\times\) on the workstation and from \(5.5\times\) to \(7\times\) on the cluster node. It is also demonstrated that solving SLAEs on GPU in the XAMG library is \(1.25\times\) faster than in the hypre library.