基于差分演化算法的双曲型方程参数识别
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摘要
差分演化算法在求解复杂优化问题时具有简单、高效的优点.本文将差分演化算法用于求解一类双曲型偏微分方程的参数识别问题,并根据所求问题的特点对算法进行了若干改进:包括基于帽子函数的参数表示和个体编码方法,用于增强算法性能的一般反向学习机制和平滑算子,以及将Tikhonov正则化和全变差正则化相结合的个体适应度计算方法.数值模拟显示,本文的算法可有效求解一维双曲型偏微分方程的参数识别问题.该算法不仅获得了高质量的近似解,而且还具有较快的收敛速度.
Differential evolution algorithm( DE) is a simple and efficient method when solving complex optimization problems. In this paper,DE was used to solve a class of parameter identification problems of hyperbolic partial differential equation,and according to the characteristics of the problems some improvements for DE were introduced. They included the parametric representation and individual coding method based on the hat function,the performance enhance schemes of the generalized opposition-based learning mechanism and the smoothing operator,and a new fitness value evaluation method which composed of the Tikhonov regularization and total variation regularization. Numerical simulations show that the proposed DE algorithm is very effective for the parameter identification problem of one-dimensional hyperbolic partial differential equation. The algorithm not only obtains the high precision solutions,but also achieves the faster convergence speed.
Differential evolution algorithm( DE) is a simple and efficient method when solving complex optimization problems. In this paper,DE was used to solve a class of parameter identification problems of hyperbolic partial differential equation,and according to the characteristics of the problems some improvements for DE were introduced. They included the parametric representation and individual coding method based on the hat function,the performance enhance schemes of the generalized opposition-based learning mechanism and the smoothing operator,and a new fitness value evaluation method which composed of the Tikhonov regularization and total variation regularization. Numerical simulations show that the proposed DE algorithm is very effective for the parameter identification problem of one-dimensional hyperbolic partial differential equation. The algorithm not only obtains the high precision solutions,but also achieves the faster convergence speed.
引文
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[17]Wang H,Rahnamayan S,Zeng S Y.General-ized opposition-based differential evolution:An experimental study[J].International Journal of Computer Applications in Technology,2012,43(4):311-319.
[18]孙志忠.偏微分方程数值解法[M].第2版.北京:科学出版社,2012:157-168.Sun Z Z.Numerical Solution of Partial Differential Equations[M].2nd Edition.Beijing:Science Press,2012:157-168(Ch).
[19]郭宝琦.确定波动方程未知系数的一个反问题[J].哈尔滨工业大学学报,1990,(4):20-26.Guo B Q.Aninverse problem for determining unknown coefficient in the wave equation[J].Journal of Harbin Institute of Technology,1990,(4):20-26(Ch).
[20]吴建成,张大力,刘家琦.一维波动方程反问题求解的正则化方法[J].计算物理,1995,12(3):415-420.Wu J C,Zhang D L,Liu J Q.The regularization methods for solving inverse problem of one-dimensional wave equation[J].Chinese Journal of Computational Physics,1995,12(3):415-420(Ch).□
[2]熊盛武,李元香,康立山,等.用演化算法求解抛物型方程扩散系数的识别问题[J].计算机学报,2000,23(3):261-265.Xiong S W,LI Y X,Kang L S,et al.Identification of spatially-varying parameters of parabolic partial differential equation by evolutionary algorithms[J].Chinese Journal of Computers,2000,23(3):261-265(Ch).
[3]熊盛武,李元香.演化参数反演方法[J].武汉大学学报(理学版),2001,47(1):37-41.Xiong S W,Li Y X.An evolutionary parameter inversion approach[J].Wuhan University Journal(Natural Science Edition),2001,47(1):37-41(Ch).
[4]Banerjee A,Abu-Mahfouz I.Evolutionary algo rithmbased parameter identification for non-linear dynamical systems[C]//IEEE Congress on Evolutionary Computation(CEC’2011).Washington,D C:IEEE,2011:1-5.
[5]Mariani V C,Neckel V J,Afonso L D,et al.Differential evolution with dynamic adaptation of mutation factor applied to inverse heat transfer problem[C]//IEEE Congress on Evolutionary Computation(CEC’2010).Washington,D C:IEEE,2010:1-6.
[6]Wang J,Wu Z J,Wang H,et al.A novel particle swarm algorithm for solving parameter identification problems on graphics hardware[J].International Journal of Computational Science and Engineering,2011,6(1):43-51.
[7]Danciu D.Numerics for hyperbolic partial differential equations(PDE)via cellular neural networks(CNN)[C]//2013 2nd International Conference on Systems and Computer Science(ICSCS).Washington,D C:IEEE,2013:183-188.
[8]彭亚绵,杨爱民,龚佃选,等.改进的最佳摄动量法及在反问题中的应用[J].数学的实践与认识,2011,41(5):186-189.Peng Y M,Yang A M,Gong D X,et al.The improved best disturbed iteration method and application of inverse problem[J].Mathematics in Practice and Theory,2011,41(5):186-189(Ch).
[9]Storn R,Price K.Differential evolution-a simple and efficient heuristic for global optim-ization over continuous space[J].Journal of Global Optimization,1997,11:341-359.
[10]彭亚绵.双曲型方程参数识别反问题的解法[J].河北理工大学学报(自然科学版),2007,29(3):105-109.Peng Y M.Numerical method for the inverse problem of parameter identified of hyperbolic[J].Journal of Hebei Polytechnic University(Natural Science Edition),2007,29(3):105-109(Ch).
[11]吴志健.演化优化及其在微分方程反问题中的应用[D].武汉:武汉大学,2004.Wu Z J.Evolutionary Optimization and Its Applications in Inverse Problems of Differential Equations[D].Wuhan:Wuhan University,2004(Ch).
[12]Rudin L I,Stanley O,Emad F.Onlinear total variation based noise removal algorithms[J].Physical D:Nonlinear Phenomena,1992,60(1):259-268.
[13]Tizhoosh H R.Opposition-based learning:A new scheme for machine intelligence[C]//Proceedings of the 2005International Conference on Computational Intelligence for Modelling,Control and Automation,and International Conference on Intelligent Agents,Web Technologies and Internet Commerce(CIMCA-IAWTIC'05).Washington,D C:IEEE,2005:695-701.
[14]Rahnamayan S,Tizhoosh H R,Salama M M A.Opposition-based differential evolution algo-rithms[C]//IEEE Congress on Evolutionary Computation(CEC’2006).Washington D C:IEEE,2006:2010-2017.
[15]Rahnamayan S,Tizhoosh H R,Salama M.Opposition versus randomness in soft computing techniques[J].Applied Soft Computing,2008,8(2):906-918.
[16]Wang H,Wu Z,Liu Y,et al.Space transform-ation search:A new evolutionary technique[C]//Proceedings of the first ACM/SIGEVO Summit on Genetic and Evolutionary Computation.New York:ACM,2009:537-544.
[17]Wang H,Rahnamayan S,Zeng S Y.General-ized opposition-based differential evolution:An experimental study[J].International Journal of Computer Applications in Technology,2012,43(4):311-319.
[18]孙志忠.偏微分方程数值解法[M].第2版.北京:科学出版社,2012:157-168.Sun Z Z.Numerical Solution of Partial Differential Equations[M].2nd Edition.Beijing:Science Press,2012:157-168(Ch).
[19]郭宝琦.确定波动方程未知系数的一个反问题[J].哈尔滨工业大学学报,1990,(4):20-26.Guo B Q.Aninverse problem for determining unknown coefficient in the wave equation[J].Journal of Harbin Institute of Technology,1990,(4):20-26(Ch).
[20]吴建成,张大力,刘家琦.一维波动方程反问题求解的正则化方法[J].计算物理,1995,12(3):415-420.Wu J C,Zhang D L,Liu J Q.The regularization methods for solving inverse problem of one-dimensional wave equation[J].Chinese Journal of Computational Physics,1995,12(3):415-420(Ch).□