并行多层快速多极子的高效预条件技术
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摘要
多层快速多极子法是基于矩量法的快速算法,具有较低的计算复杂度和存储复杂度,被广泛应用于目标电磁散射特性分析。对于复杂结构电磁目标,由于矩阵条件数较差,往往存在迭代收敛慢甚至不收敛的问题。针对这一情况,文中利用快速多极子的近区矩阵,结合稀疏矩阵方程求解构造了一种高效预条件。数值实例表明该方法相比于块对角预条件效果更好,能有效加速多层快速多极子迭代过程。
Based on the method of moment(MoM), the multilevel fast multipole algorithm(MLFMA) has the property of low computation and storage complexity, and it is widely used in calculating electromagnetic scattering problems. However, for a target with complex structure, there will be an ill-conditioned matrix, making the iteration time too long or the problem non-convergence. Considering such situation, we propose a kind of preconditioner combining the near-field matrix and sparse matrix solver for MLFMA. Compared with the block diagonal preconditioned methods, this method has better convergence characteristic.
Based on the method of moment(MoM), the multilevel fast multipole algorithm(MLFMA) has the property of low computation and storage complexity, and it is widely used in calculating electromagnetic scattering problems. However, for a target with complex structure, there will be an ill-conditioned matrix, making the iteration time too long or the problem non-convergence. Considering such situation, we propose a kind of preconditioner combining the near-field matrix and sparse matrix solver for MLFMA. Compared with the block diagonal preconditioned methods, this method has better convergence characteristic.
引文
[1] 张玉,赵勋旺,陈岩,等.计算电磁学中的超大规模并行矩量法[M].西安:西安电子科技大学出版社,2016 Zhang Y,Zhao X W,Chen Y,et al.Superscale parallel moment method in computational electromagnetism[M].Xi'an:Press of Xi'an University of Electronic Science and Technology,2016
[2] Song J M,Chew W C.Multilevel fast multipole algorithm for solving combined field integral equation of electromagnetic scattering[J].Microwave & Optical Technology Letters,2010,10(1):14-19
[3] Hestenes M R,Stiefel E.Methods of conjugate gradients for solving linear systems[J].J.res.nat.bur.stand,1952,49(6):409-436
[4] Saad Y,Schultz M H.GMRES:A generalized minimal residual algorithm for solving non-symmetric linear systems[J].Society for Industrial and Applied Mathematics,1986,7(3):856-869
[5] van der Vorst H A.Bi-CGSTAB:A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems[J].SIAM Journal on Scientific and Statistical Computing,1992,13(2):631-644
[6] Heldring A,Rius J M,Ligthart L.New block ILU preconditioner scheme for numerical analysis of very large electromagnetic problems[J].IEEE Transactions on Magnetics,2002,38(2):337-340
[7] Alleon G,Benzi M,Giraud L.Sparse approximate inverse preconditioning for dense linear systems[J].Arising in Computational Electromagnetics,Numerical Algorithms,1997(16):1-15
[8] 项铁铭,梁昌洪.一种新型针对快速多极子法(FMM)的预条件技术[J].微波学报,2004,20(1):67-70 Xiang T M,Liang C H.A new preconditioner for FMM implementation[J].Journal of Microwaves,2004,20(1):67-70
[9] Bruun A I O H.Direct methods for sparse matrices[J].Mathematics of Computation,1980,9(123):874
[2] Song J M,Chew W C.Multilevel fast multipole algorithm for solving combined field integral equation of electromagnetic scattering[J].Microwave & Optical Technology Letters,2010,10(1):14-19
[3] Hestenes M R,Stiefel E.Methods of conjugate gradients for solving linear systems[J].J.res.nat.bur.stand,1952,49(6):409-436
[4] Saad Y,Schultz M H.GMRES:A generalized minimal residual algorithm for solving non-symmetric linear systems[J].Society for Industrial and Applied Mathematics,1986,7(3):856-869
[5] van der Vorst H A.Bi-CGSTAB:A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems[J].SIAM Journal on Scientific and Statistical Computing,1992,13(2):631-644
[6] Heldring A,Rius J M,Ligthart L.New block ILU preconditioner scheme for numerical analysis of very large electromagnetic problems[J].IEEE Transactions on Magnetics,2002,38(2):337-340
[7] Alleon G,Benzi M,Giraud L.Sparse approximate inverse preconditioning for dense linear systems[J].Arising in Computational Electromagnetics,Numerical Algorithms,1997(16):1-15
[8] 项铁铭,梁昌洪.一种新型针对快速多极子法(FMM)的预条件技术[J].微波学报,2004,20(1):67-70 Xiang T M,Liang C H.A new preconditioner for FMM implementation[J].Journal of Microwaves,2004,20(1):67-70
[9] Bruun A I O H.Direct methods for sparse matrices[J].Mathematics of Computation,1980,9(123):874