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<p>In this paper, we propose two mixed precision algorithms for block-Jacobi preconditioner (BJAC): a fixed low precision strategy and an adaptive precision strategy. We evaluate the performance improvement of the proposed mixed precision BJAC preconditioners combined with the preconditioned conjugate gradient (PCG) method using problems including diffusion equations and radiation hydrodynamics equations. Numerical results show that, compared with the uniform high precision PCG, the mixed precision preconditioners can achieve speedups from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42514_2025_213_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(1.3\times\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.3</mn> <mo>×</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42514_2025_213_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(1.8\times\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.8</mn> <mo>×</mo> </mrow> </math></EquationSource> </InlineEquation> without losing accuracy. Furthermore, we observe the phenomenon of convergence delay in some test cases for the mixed precision preconditioners, and analyse the correlation between matrix features and convergence delay behaviors. Some interesting conclusions are obtained which are significant and valuable for the design of more efficient mixed precision preconditioners.</p>

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Mixed precision block-Jacobi preconditioner: algorithms, performance evaluation and feature analysis

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Ningxi Tian,
  • Silu Huang,
  • Xiaowen Xu

摘要

In this paper, we propose two mixed precision algorithms for block-Jacobi preconditioner (BJAC): a fixed low precision strategy and an adaptive precision strategy. We evaluate the performance improvement of the proposed mixed precision BJAC preconditioners combined with the preconditioned conjugate gradient (PCG) method using problems including diffusion equations and radiation hydrodynamics equations. Numerical results show that, compared with the uniform high precision PCG, the mixed precision preconditioners can achieve speedups from \(1.3\times\) 1.3 × to \(1.8\times\) 1.8 × without losing accuracy. Furthermore, we observe the phenomenon of convergence delay in some test cases for the mixed precision preconditioners, and analyse the correlation between matrix features and convergence delay behaviors. Some interesting conclusions are obtained which are significant and valuable for the design of more efficient mixed precision preconditioners.