一类不适定问题的粒子群算法求解
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摘要
针对测量中的不适定问题,提出了一种基于粒子群算法的优化求解方法。将不适定问题通过Tikhonov正则化,建立优化目标函数,将方程组的求解转化为无约束优化问题。利用L-曲线法确定正则参数,取优化目标函数为粒子群算法的适应度函数,通过改进的变异粒子群算法进行随机搜索最优解。最后通过两个测量中的不适定问题的数值算例,利用粒子群算法进行搜索求解,相较于一般的正则化解法,该方法求得的结果更接近真值。
Aiming at the ill-posed problem in measurement,an optimization method based on particle swarm optimization( PSO) is proposed. The ill-posed problem is regularized by Tikhonov,and the optimization objective function is established. The solution of the equations is transformed into an unconstrained optimization problem. The L-curve method is used to determine the regularization parameters,and the optimization objective function is taken as the fitness function of the particle swarm optimization algorithm. The improved mutation particle swarm optimization algorithm is used to search the optimal solution randomly. Finally,through two numerical examples of ill-posed problems in measurement,the particle swarm optimization algorithm is used to search for solutions. Compared with the general regularization method,the results obtained by this method are closer to the true value.
Aiming at the ill-posed problem in measurement,an optimization method based on particle swarm optimization( PSO) is proposed. The ill-posed problem is regularized by Tikhonov,and the optimization objective function is established. The solution of the equations is transformed into an unconstrained optimization problem. The L-curve method is used to determine the regularization parameters,and the optimization objective function is taken as the fitness function of the particle swarm optimization algorithm. The improved mutation particle swarm optimization algorithm is used to search the optimal solution randomly. Finally,through two numerical examples of ill-posed problems in measurement,the particle swarm optimization algorithm is used to search for solutions. Compared with the general regularization method,the results obtained by this method are closer to the true value.
引文
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[2]李功胜,马逸尘.应用正则化子建立求解不适定问题的正则化方法的探讨[J].数学进展,2000,29(6):531-541.
[3]沈云中,许厚泽.不适定方程正则化解法的谱分解式[J].大地测量与地球动力学,2002,22(3):10-14.
[4]王新洲,刘丁酉.最小二乘估计中法方程的迭代解法[J].湖北民族学院学报自然科学版,2002,20(3):1-4.
[5]唐丽,李鹏飞.主元加权迭代法求解病态线性方程组[J].科学技术与工程,2012,12(2):381-383.
[6]王倩,戴华.求解离散不适定问题的正则化GMERR方法[J].计算数学,2013,35(2):195-204.
[7]NeumanA,Reichel L,Sadok H. Implementations of range restricted iterative methods for discrete ill-posed problems[J]. Linear Algebra Appl,2012,436:3974-3990.
[8]潘美芹,贺国平.基于共轭梯度法的函数优化混合遗传算法[J].山东科技大学学报(自然科学版),2000,19(4):10-13.
[9]陈内萍.一种解病态线性方程组的神经网络算法[J].湖南师范大学自然科学报,2007,30(3):38-41.
[10]李海滨,尚凡华.基于神经网络的病态线性方程组求解[J].辽宁工程技术大学学报,2007,26(6):956-958.
[11]黄松奇,黄守佳.用遗传算法求解病态线性方程组[J].数学的实践与认识,2003,33(8):97-100.
[12]白俊雨,谢爽,梁义强.基于蚁群算法的Tikhonov正则参数优化计算[J].新疆石油天然气,2017,13(1):30-36.
[13]周利军,彭卫,邹芳,等.自适应变异粒子群算法[J].计算机工程与应用,2016,52(7):50-56.
[14]刘志刚,曾嘉俊,韩志伟.基于个体最优位置的自适应变异扰动粒子群算法[J].西南交通大学学报. 2012,47(5):761-768.
[15]Zhang Wen,LiuYutian,Clerc M. An adaptive PSO algorithm for reactive power optimization[C]//Proc of the 6th International Conference on Advanced in Power System Control,Operation and Management,Piscataway,NJ:IEEE Press,2003:302-307.
[16]王振杰.测量中不适定问题的正则化解法[M].北京:科学出版社,2006:86-92.