Gauss-Seidel算法作为线性方程组的求解器,在并行计算领域具有广泛应用,而面向异构众核架构开发其细粒度并行性一直是具有挑战性的问题.针对非结构网格问题,基于代数分块并行思路提出了面向异构众核架构的块Gauss-Seidel/Jacobi算法,将...Gauss-Seidel算法作为线性方程组的求解器,在并行计算领域具有广泛应用,而面向异构众核架构开发其细粒度并行性一直是具有挑战性的问题.针对非结构网格问题,基于代数分块并行思路提出了面向异构众核架构的块Gauss-Seidel/Jacobi算法,将其作为区域分解算法的子区域求解器.面向神威太湖之光超级计算机的异构众核架构,设计并实现了该算法.为充分利用神威太湖之光国产SW26010芯片中每个CPE拥有的高速LDM(Local Data Memory),缓解通信瓶颈,设计了多行块通信打包、计算与通信重叠性能优化策略和丢弃非关键元素的低通信复杂性数值优化方法.数值实验结果显示,相较于串行Gauss-Seidel算法,优化后的块Gauss-Seidel/Jacobi算法预处理过程加速比最高可达到4.16倍.以1040核的测试数据为基准,在处理器核数达到33280时,块Gauss-Seidel/Jacobi预条件算法的并行效率达到61%.展开更多
In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real ...In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real elements). Now the new matrix gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.展开更多
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 m...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×to 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.展开更多
文摘Gauss-Seidel算法作为线性方程组的求解器,在并行计算领域具有广泛应用,而面向异构众核架构开发其细粒度并行性一直是具有挑战性的问题.针对非结构网格问题,基于代数分块并行思路提出了面向异构众核架构的块Gauss-Seidel/Jacobi算法,将其作为区域分解算法的子区域求解器.面向神威太湖之光超级计算机的异构众核架构,设计并实现了该算法.为充分利用神威太湖之光国产SW26010芯片中每个CPE拥有的高速LDM(Local Data Memory),缓解通信瓶颈,设计了多行块通信打包、计算与通信重叠性能优化策略和丢弃非关键元素的低通信复杂性数值优化方法.数值实验结果显示,相较于串行Gauss-Seidel算法,优化后的块Gauss-Seidel/Jacobi算法预处理过程加速比最高可达到4.16倍.以1040核的测试数据为基准,在处理器核数达到33280时,块Gauss-Seidel/Jacobi预条件算法的并行效率达到61%.
文摘In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real elements). Now the new matrix gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.
基金National Key Research and Development Program of China(No.2023YFB3001605).
文摘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×to 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.
