)不同预处理效果 P=(D-L)D-1(D-L) P2=TriDiag(A) P3 ichol(A) P4 michol(A) n 82 162 322 642 (A) 32.2 166 441 1712 K(PA) 4.89 15.5 56.3 216 K(P2A) 16.6 58.7 221 856 K(P3A) 3.69 11.1 39.8 152 K(PA) 3.00 6.65 16.5 43.5 http://aath.ecnu.edu.cn/-jypan 6/72
不同预处理效果 P1 = (D − L)D −1 (D − L ⊺ ) P2 = TriDiag(A) P3 = ichol(A) P4 = michol(A) n 8 2 162 322 642 κ(A) 32.2 166 441 1712 κ(P −1 1 A) 4.89 15.5 56.3 216 κ(P −1 2 A) 16.6 58.7 221 856 κ(P −1 3 A) 3.69 11.1 39.8 152 κ(P −1 4 A) 3.00 6.65 16.5 43.5 http://math.ecnu.edu.cn/~jypan 6/72
PCG convergence history 100 105 2装5393oc00909e0909903e0ee3030e0ne9e09e60406e为 10-10 CG SGS 1015 -TriDiag ichol -michol 10-20 10 20 30 40 50 60 70 80 90 100 Performance of different preconditioners for 2D Poisson with n=642 http:/aath.ecn.edu.cm/jp 7/72
10 20 30 40 50 60 70 80 90 100 10-20 10-15 10-10 10-5 100 PCG convergence history CG SGS TriDiag ichol michol Performance of different preconditioners for 2D Poisson with n = 642 http://math.ecnu.edu.cn/~jypan 7/72
预处理举例：分数阶扩散方程 「-D[(oD-8-(1-)D）国=国， x∈0,1，K(a=1+8x,3=0.5,0=0.5 Convergence history for N=4096 Condition number 10 一GMRES n cond(A) 28 5191 10 29 14813 210 42143 211 119653 102 212 339259 213 961081 10 0 200 400 00 80 1000 http://aath.ecnu.edu.cn/-jypan 8/72
预处理举例: 分数阶扩散方程 − D [ K(x) ( θ 0 D 1−β x −(1 − θ) x D 1−β 1 ) u(x) ] = f(x), x ∈ [0, 1], K(x) = 1 + 8x, β = 0.5, θ = 0.5. Condition number n cond(A) 2 8 5191 2 9 14813 2 10 42143 2 11 119653 2 12 339259 2 13 961081 Convergence history for N = 4096 http://math.ecnu.edu.cn/~jypan 8/72
简单循环预处理子 Convergence history after preconditioning Condition number 10° -Preconditioned GMRES cond(A) cond(P-1A) 102 28 10 5191 76 108 29 14813 112 210 109 42143 162 100 21 119653 235 102 212 339259 337 104 213 961081 482 106 0-00900 10 20 3040 506070 80 http://math.ecnu.edu.cn/-jypan 9/72
简单循环预处理子 Condition number n cond(A) cond(P −1A) 2 8 5191 76 2 9 14813 112 2 10 42143 162 2 11 119653 235 2 12 339259 337 2 13 961081 482 Convergence history after preconditioning http://math.ecnu.edu.cn/~jypan 9/72
关于预处理方法的相关参考资料 Saad,Iterative Methods for Sparse Linear Systems,2nd edition,2003. Chen,Matrir Preconditioning Techniques and Applications,2005. Malek and Z.Strakos,Preconditioning and the Conjugate Gradient Method in The Contert of Solving PDEs,2015. Bertaccini and Durastante,Iterative Methods and Preconditioning for Large and Sparse Linear Systems With Applications,2018. 0谷同祥等，迭代方法和预处理技术（下），科学出版社，2015. Benzi,Preconditioning techniques for large linear systems:A survey,JCP,2002, 418-477. Mardal and Winther,Preconditioning discretizations of systems of partial differ- ential equations,NLAA,2011,1-40. Wathen,Preconditioning,Acta Numerica,2015,329-376 DPearson and Pestana,Preconditioners for Krylov subspace methods:An overview, GAMM,2020. http://aath.ecnu.edu.cn/-jypan 10/72
关于预处理方法的相关参考资料 Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, 2003. Chen, Matrix Preconditioning Techniques and Applications, 2005. Málek and Z. Strakoˇs, Preconditioning and the Conjugate Gradient Method in The Context of Solving PDEs, 2015. Bertaccini and Durastante, Iterative Methods and Preconditioning for Large and Sparse Linear Systems With Applications, 2018. 谷同祥等, 迭代方法和预处理技术 (下), 科学出版社, 2015. ▷ Benzi, Preconditioning techniques for large linear systems: A survey, JCP, 2002, 418–477. ▷ Mardal and Winther, Preconditioning discretizations of systems of partial differential equations, NLAA, 2011, 1–40. ▷ Wathen, Preconditioning, Acta Numerica, 2015, 329–376. ▷ Pearson and Pestana, Preconditioners for Krylov subspace methods: An overview, GAMM, 2020. http://math.ecnu.edu.cn/~jypan 10/72