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针对二维泊松方程在实际应用过程中几种常用方法存在计算量大、易发散、局部收敛等不足, 提出了一种改进算法.该算法基于并行超松弛迭代法,采用遗传算法对松弛因子进行全局寻优, 解决了超松弛迭代法求解泊松方程时最佳松弛因子难以确定的问题. 构建了多目标适应度函数,优化了遗传算子参数,分析了算法的计算量、计算时间与误差精度, 与传统方法进行了对比研究.结果表明:松弛因子对泊松方程求解的速度与精度影响显著; 改进算法能减少迭代次数,节省计算时间,加快方程的求解;算法适合于求解计算量较大、 精度要求较高的时域有限差分方程,而且精度要求越高,算法的性能越好,节省的时间也越多.There exist some disadvantages in the calculation of two-dimensional Poisson equation with several common methods. A new ameliorative algorithm is presented. It is based on a parallel successive over-relaxation (PSOR) method, by using the multi-objective genetic algorithm to search for optimal relaxation factor, with which the problem of optimal relaxation factor selection in PSOR is solved. The multi-objective fitness function is constructed, with which the genetic algorithm parameters are optimized. The analysis mainly focuses on algorithm computation, time cost and accuracy of error correction. The performance of the ameliorative algorithm is compared with those of Jacobi, Gauss-Seidel, Successive over relaxation iteration (SOR) and PSOR. Experimental results show that relaxation factor has a significant effect on the speed of solving Poisson equation, as well as the accuracy. The improved algorithm can increase the speed of iteration and obtain higher accuracy than traditional algorithm. It is suited for solving complicated finite difference time domain equations which need high accuracy. The higher the accuracy requirement, the better the performance of the algorithm is and the more computation time can also be saved.
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Keywords:
- poisson equation /
- genetic algorithm /
- parallel successive over-relaxation method /
- finite difference method
[1] Wang X Y, Zhang H M, Wang G Y, Song J J, Qin S S, Qu J T 2011 Acta Phys. Sin. 60 027102 (in Chinese) [王晓艳, 张鹤鸣, 王冠宇, 宋建军, 秦珊珊, 屈江涛 2011 60 027102]
[2] Shang Y, Huo B Z, Meng C N, Yuan J H 2010 Acta Phys. Sin. 59 8178 (in Chinese) [尚英, 霍丙忠, 孟春宁, 袁景和 2010 59 8178]
[3] Ji F Y, Zhang S L 2012 Acta Phys. Sin. 61 080202 (in Chinese) [吉飞宇, 张顺利 2012 61 080202]
[4] Ma J W, Yang H Z, Zhu Y P 2001 Acta Phys. Sin. 50 1415 (in Chinese) [马坚伟, 杨慧珠, 朱亚平 2001 50 1415]
[5] Liu S K, Fu Z T, Liu S D 2001 Phys. Lett. A 289 69
[6] Kohno T, Kotakemori H, Nikia H 1997 Linear Algebra Appl. 267 113
[7] Hadjidimos A 2000 Journal of Computational and Applied Mathematics 123 77
[8] Smith B F, Bjorstad P E, Gropp W D 1996 Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge: Cambridge University Press) p124
[9] Wang X B, Liang Z C, Wu Z S, 2012 Acta Phys. Sin. 61 124104 (in Chinese) [王晓冰, 梁子长, 吴振森 2012 61 124104]
[10] He J, Xu J Y, Yao X 2000 IEEE Trans on Evolutionary Computation 4 295
[11] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 51 2459]
[12] Zhao Z J, Zhen S L, Shang J N, Kong X Z 2007 Acta Phys. Sin. 56 6760 (in Chinese) [赵知劲, 郑仕链, 尚俊娜, 孔宪正 2007 56 6760]
[13] Dutta D, Dutta P, Sil J 2012 Proceedings of the 1st International Conference on Recent Advances in Information Technology, Dhanbad, India, March 15-17 2012 p548
[14] Sweilam N H, Moharram H M, Ahmed S 2012 Proceedings of the 8th International Conference on Informatics and Systems, Cairo, Egypt, May 14-16, 2012 p78
[15] Xu Q Y 2011 Proceedings of the 2011 Inernational Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery, Beijing, China, Octorber 10-12, 2011 p295
[16] Wang B Z 2002 Computational electromagnetic (Beijing: Science Press) p34 (in Chinese) [王秉中 2002 计算电磁学(北京:科学出版社) 第34页]
[17] Srinivas M, Patnaik L M 1994 IEEE Trans. on SMC 24 656
[18] Xie Z C, Zhou Y Q 2009 Mathematics in Practice and Theory 39 154 (in Chinese) [谢竹诚, 周永权 2009 数学的实践与认知 39 154]
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[1] Wang X Y, Zhang H M, Wang G Y, Song J J, Qin S S, Qu J T 2011 Acta Phys. Sin. 60 027102 (in Chinese) [王晓艳, 张鹤鸣, 王冠宇, 宋建军, 秦珊珊, 屈江涛 2011 60 027102]
[2] Shang Y, Huo B Z, Meng C N, Yuan J H 2010 Acta Phys. Sin. 59 8178 (in Chinese) [尚英, 霍丙忠, 孟春宁, 袁景和 2010 59 8178]
[3] Ji F Y, Zhang S L 2012 Acta Phys. Sin. 61 080202 (in Chinese) [吉飞宇, 张顺利 2012 61 080202]
[4] Ma J W, Yang H Z, Zhu Y P 2001 Acta Phys. Sin. 50 1415 (in Chinese) [马坚伟, 杨慧珠, 朱亚平 2001 50 1415]
[5] Liu S K, Fu Z T, Liu S D 2001 Phys. Lett. A 289 69
[6] Kohno T, Kotakemori H, Nikia H 1997 Linear Algebra Appl. 267 113
[7] Hadjidimos A 2000 Journal of Computational and Applied Mathematics 123 77
[8] Smith B F, Bjorstad P E, Gropp W D 1996 Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge: Cambridge University Press) p124
[9] Wang X B, Liang Z C, Wu Z S, 2012 Acta Phys. Sin. 61 124104 (in Chinese) [王晓冰, 梁子长, 吴振森 2012 61 124104]
[10] He J, Xu J Y, Yao X 2000 IEEE Trans on Evolutionary Computation 4 295
[11] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 51 2459]
[12] Zhao Z J, Zhen S L, Shang J N, Kong X Z 2007 Acta Phys. Sin. 56 6760 (in Chinese) [赵知劲, 郑仕链, 尚俊娜, 孔宪正 2007 56 6760]
[13] Dutta D, Dutta P, Sil J 2012 Proceedings of the 1st International Conference on Recent Advances in Information Technology, Dhanbad, India, March 15-17 2012 p548
[14] Sweilam N H, Moharram H M, Ahmed S 2012 Proceedings of the 8th International Conference on Informatics and Systems, Cairo, Egypt, May 14-16, 2012 p78
[15] Xu Q Y 2011 Proceedings of the 2011 Inernational Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery, Beijing, China, Octorber 10-12, 2011 p295
[16] Wang B Z 2002 Computational electromagnetic (Beijing: Science Press) p34 (in Chinese) [王秉中 2002 计算电磁学(北京:科学出版社) 第34页]
[17] Srinivas M, Patnaik L M 1994 IEEE Trans. on SMC 24 656
[18] Xie Z C, Zhou Y Q 2009 Mathematics in Practice and Theory 39 154 (in Chinese) [谢竹诚, 周永权 2009 数学的实践与认知 39 154]
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