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Mixed Precision Block-Jacobi Preconditioner: Algorithms, Performance Evaluation and Feature Analysis

Ningxi Tian, Silu Huang, Xiaowen Xu

TL;DR

The performance improvement of the proposed mixed precision BJAC preconditioners combined with the preconditioned conjugate gradient (PCG) method is evaluated using problems including diffusion equations and radiation hydrodynamics equations.

Abstract

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 algorithm using problems including diffusion equations and radiation hydrodynamics equations. Numerical results show that, compared to the uniform high precision PCG algorithm, the mixed precision preconditioners can achieve speedups from 1.3 to 1.8 without sacrificing accuracy. Furthermore, we observe the phenomenon of convergence delay in some test cases for the mixed precision preconditioners, and further analyse the matrix features associate with the convergence delay behavior.

Mixed Precision Block-Jacobi Preconditioner: Algorithms, Performance Evaluation and Feature Analysis

TL;DR

The performance improvement of the proposed mixed precision BJAC preconditioners combined with the preconditioned conjugate gradient (PCG) method is evaluated using problems including diffusion equations and radiation hydrodynamics equations.

Abstract

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 algorithm using problems including diffusion equations and radiation hydrodynamics equations. Numerical results show that, compared to the uniform high precision PCG algorithm, the mixed precision preconditioners can achieve speedups from 1.3 to 1.8 without sacrificing accuracy. Furthermore, we observe the phenomenon of convergence delay in some test cases for the mixed precision preconditioners, and further analyse the matrix features associate with the convergence delay behavior.
Paper Structure (20 sections, 7 equations, 8 figures, 4 tables, 3 algorithms)

This paper contains 20 sections, 7 equations, 8 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mixed Precision Block-Jacobi Preconditioner
  4. The Block-Jacobi Preconditioner(BJAC)
  5. Mixed Precision Block-Jacobi Preconditioner(MP-BJAC)
  6. Fixed Low Precision BJAC Preconditioner
  7. Adaptive Precision BJAC Preconditioner
  8. Performance Evualation and Feature Analysis
  9. Test Problems
  10. 3D Diffusion Equation (Diff3D)
  11. 3D Radiation Hydrodynamic Equations (RHD3D)
  12. Test Setting
  13. Performance Evaluation
  14. Convergence Behavior Analysis
  15. Convergence Delays Phenomenon
  16. ...and 5 more sections

Figures (8)

  • Figure 1: Time and speedup of mixed precision BJAC PCG compared to fp64-uniform-BJAC PCG: Diff3D problems.
  • Figure 2: Time and speedup of mixed precision BJAC PCG compared to fp80-uniform-BJAC PCG: RHD3D problems.
  • Figure 3: Iteration numbers for fp64-uniform- and mixed precision BJAC: Diff3D problems.
  • Figure 4: Iteration numbers for fp80-uniform- and mixed precision BJAC: RHD3D problems.
  • Figure 5: RelResNorm curves of the aMP-BJAC(hl) and aMP-BJAC(lh) for solving the RHD3D-1T problem with three typical values of ${adp}_{tol}$. Upper panel: the overall result. Lower panel: the detailed result after the 500th iteration.
  • ...and 3 more figures