求解多尺度稀疏矩阵的代数界面优先AMG光滑子

AN AMG SMOOTHER BASED ON ALGEBRAIC INTERFACE-PRIOR FOR SOLVING MULTL-SCALE SPARSE MATRICES

光滑子是影响代数多重网格算法(AMG)求解效率的重要组件之一.本文考虑实际应用中普遍出现的一类多尺度稀疏矩阵,由于多尺度性质的影响,现有AMG光滑子的光滑效果不理想,从而影响AMG算法求解该类方程的效率.借助代数界面的概念,本文分析了代数界面对松弛型光滑子的影响,并通过扩展代数界面的内涵,设计了一种代数界面优先的光滑子(AI-Smoother).以Gauss-Seidel(GS)光滑子为例,通过三维模型问题和实际问题测试了该光滑子(AI-GS)的有效性.测试表明,与自然序GS光滑子相比,AI-GS有效改善了AMG算法的收敛速度.对于三维随机系数扩散方程百万自由度算例,AI-GS可获得28.2%的加速,对于激光聚变应用中的三温方程百万自由度算例,AI-GS可获得28.8%的加速.

The smoother is one of the important components that affects the solution efficiency of the Algebraic Multigrid Algorithm(AMG).This paper considers a class of multi-scale sparse matrices that commonly appear in practical applications.Due to the influence of multi-scale properties,the smoothing effect of the existing AMG smoothers is not work well,which affects the efficiency of the AMG algorithm for solving multi-scale sparse matrices.Using the concept of algebraic interface,this paper analyzes the influence of algebraic interface on relaxation smoothers,and by extending the concept of algebraic interface,an algebraic interface-prior smoother(AI-Smoother)is designed.Taking Gauss-Seidel(GS)smoother as an example,the effectiveness of the AI-porior smoother(AI-GS)is tested through 3D model problems and practical problems.Numerical results show that,compared with the original GS smoother,AI-GS effectively improves the convergence speed of the AMG algorithm.AI-GS can achieve a 28.2% speedup for the three-dimensional random coefficient diffusion equation with one million degrees of freedom.For the three-temperature equation in laser fusion applications,AI-GS can achieve a 28.8% speedup.