基于贝叶斯-微分进化算法的污染源识别反问题
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摘要
针对污染源瞬时排放的河流水污染事件反问题,通过贝叶斯统计方法和二维水质对流-扩散方程,建立水体污染识别模型,得到关于污染源强度、污染源位置和污染源排放时刻3个未知参数的后验概率密度函数。运用最大似然估计的思想,采用微分进化算法,求解使后验概率密度函数达到最大值的参数,作为模型未知参数的估计值。算例表明:运用贝叶斯微分进化算法,3个未知参数估计值迭代50次时可以达到稳定,当迭代次数达到280次时,可与真值完全重合;与贝叶斯-蒙特卡洛法相比,贝叶斯-微分进化算法可使3个未知参数估计值达到稳定时的迭代次数降低97.5%,均值误差分别减少1.69%、2.12%和4.03%,具有收敛快、精度高的特点。
Aiming at the inverse problem of water pollution of river with pollutant source discharged instantaneously,a methodical model was constructed based on Bayesian statistical method and two-dimensional water quality model. The posterior probability distribution of unknown parameters including source's position,intensity and discharging time was deduced. The parameters made the posterior probability density function reach the maximum value by the idea of maximum likelihood estimate and differential evolution algorithm( DE),which were viewed as the estimates of model parameters. The example showed that three estimate parameters could reach the stable state based on Bayesian-DE after 50 iterations,and correspond with truth values after 280 iterations. Compared with Bayesian-Markov chain Monte Carlo simulation( Bayesian-MCMC),97. 5% iterations of three estimate parameters reaching the stable state could be reduced,and the mean errors were decreased by 1. 69%,2. 12% and 4. 03% through the use of BayesianDE featuring rapid convergence and high accuracy.
Aiming at the inverse problem of water pollution of river with pollutant source discharged instantaneously,a methodical model was constructed based on Bayesian statistical method and two-dimensional water quality model. The posterior probability distribution of unknown parameters including source's position,intensity and discharging time was deduced. The parameters made the posterior probability density function reach the maximum value by the idea of maximum likelihood estimate and differential evolution algorithm( DE),which were viewed as the estimates of model parameters. The example showed that three estimate parameters could reach the stable state based on Bayesian-DE after 50 iterations,and correspond with truth values after 280 iterations. Compared with Bayesian-Markov chain Monte Carlo simulation( Bayesian-MCMC),97. 5% iterations of three estimate parameters reaching the stable state could be reduced,and the mean errors were decreased by 1. 69%,2. 12% and 4. 03% through the use of BayesianDE featuring rapid convergence and high accuracy.
引文
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[22]杨海东,肖宜,王卓民,等.突发性水污染事件溯源方法[J].水科学进展,2014,25(1):122-128.YANG Haidong,XIAO Yi,WANG Zhuomin,et al.On source identification method for sudden water pollution accidents[J].Advances in Water Science,2014,25(1):122-128.
[23]闵涛,周孝德,张世梅,等.对流-扩散方程源项识别反问题的遗传算法[J].水动力学研究与进展,A辑,2004,19(4):520-524.MIN Tao,ZHOU Xiaode,ZHANG Shimei,et al.Genetic algorithm to an inverse problem of source term identification for convection-diffusion equation[J].Journal of Hydrodynamics(Ser.A),2004,19(4):520-524.
[24]牟行洋.基于微分进化算法的污染物源项识别反问题研究[J].水动力学研究与进展(A辑),2011,26(1):24-30.MOU Xingyang.Research on inverse problem of pollution source term identification based on differential evolution algorithm[J].Journal of Hydrodynamics(Ser.A),2011,26(1):24-30.
[25]郑春苗,BENNETT G D.地下水污染物迁移模拟(第二版)[M].北京:高等教育出版社,2009:70-79.
[2]JHA M,DATTA B.Three-dimensional groundwater contamination source identification using adaptive simulated annealing[J].Journal of Hydrologic Engineering,2012,18(3):307-317.
[3]HOLLAND J.Adaptation in natural and artificial systems[M].Ann Arbor:University of Michigan Press,1975.
[4]RUZEK B,KVASNICKA M.Differential evolution algorithm in the earthquake hypocenter location[J].Pure and Applied Geophysics,2001,158(4):667-693.
[5]赵光权.基于贪婪策略的微分进化算法及其应用研究[D].哈尔滨:哈尔滨工业大学,2007.ZHAO Guangquan.Differential evolution algorithm with greedy strategy and its applications[D].Harbin:HarbinInstitute of Technology,2007.
[6]苏海军,杨煜普,王宇嘉.微分进化算法的研究综述[J].系统工程与电子技术,2008,30(9):1793-1797.SU Haijun,YANG Yupu,WANG Yujia.Research on differential evolution algorithm:a survey[J].Systems Engineering and Electronics,2008,30(9):1793-1797.
[7]STRON R,PRICE K.Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces[J].Journal of Global Optimization,1997,11(4):341-359.
[8]ABBASS H A.The self-adaptive Pareto differential evolution algorithm[C]//Proceedings of 2002 IEEE Congress on Evolutionary Computation.Honolulu,USA:IEEE,2002:831-836.
[9]STRON R,PRICE K.Minimizing the real functions of the ICEC'96 contest by differential evolution[C]//IEEE International Conference on Evolutionary Computation.Nagoya,Japan:IEEE,1996:842-844.
[10]BECERRA R,COELLO C.Cultured differential evolution for constrained optimization[J].Computer Methods in Applied Mechanics and Engineering,2006,195(33-36):4303-4322.
[11]BREST J,GREINER S,BOSKOVIC B,et al.Selfadapting control parameters in differential evolution:a comparative study on numerical benchmark problems[J].IEEE Transaction on Evolutionary Computation,2006,10(6):646-657.
[12]FAN H,LAMPINEN J.A trigonometric mutation operation to differential evolution[J].Journal of Global Optimization,2003,27(1):105-129.
[13]CHAKRABORY U,DAS S,KONAR A.Differential evolution with local neighborhood[C]//IEEE Congress on Evolutionary Computation.Vancouver,Canada:IEEE,2006:2042-2049.
[14]郝竹林,冯民权,闵涛,等.地下水渗流参数反演的微分进化算法[J].自然灾害学报,2015,24(1):55-60.HAO Zhulin,FENG Minquan,MIN Tao,et al.Differential evolution algorithm for parameter inversion of groundwater seepage flow[J].Journal of Natural Disasters,2015,24(1):55-60.
[15]刘英霞,王希常,唐晓丽,等.基于小波域特征和贝叶斯估计的目标检测算法[J].山东大学学报(工学版),2017,47(2):63-70.LIU Yingxia,WANG Xichang,TANG Xiaoli,et al.Object detection algorithm based on Bayesian probability estimation in wavelet domain[J].Journal of Shandong University(Engineering Sciences),2017,47(2):63-70.
[16]HAARIO H,SAKSMAN E,TAMMINEN J.An adaptive metropolis algorithm[J].Bernoulli,2001,7(2):223-242.
[17]ZENG L,SHI L,ZHANG D,et al.A sparse grid based Bayesian method for contaminant source identification[J].Advances in Water Resources,2012,37(1):1-9.
[18]王海军,葛红娟,张圣燕.基于L1范数和最小软阈值均方的目标跟踪算法[J].山东大学学报(工学版),2016,46(3):14-22.WANG Haijun,GE Hongjuan,ZHANG Shengyan.Object tracking via L1 norm and least soft-threshold square[J].Journal of Shandong University(Engineering Sciences),2016,46(3):14-22.
[19]曹瑛,王明文,陶亮.基于Markov网络的检索模型[J].山东大学学报(理学版),2006,41(3):126-130.CAO Ying,WANG Mingwen,TAO Liang.Information retrieval model based on Markov network[J].Journal of Shandong University(Natural Science),2006,41(3):126-130.
[20]曾平,王婷.贝叶斯错误发现率[J].山东大学学报(医学版),2012,50(3):120-124.ZENG Ping,WANG Ting.Bayesian false discovery rate[J].Journal of Shandong University(Health Sciences),2012,50(3):120-124.
[21]陈海洋,腾彦果,王金生,等.基于Bayesian-MCMC方法的水体污染识别反问题[J].湖南大学学报(自然科学版),2012,39(6):74-78.CHEN Haiyang,TENG Yanguo,WANG Jinsheng,et al.Event source identification of water pollution based on Bayesian-MCMC[J].Journal of Hunan University(Natural Sciences),2012,39(6):74-78.
[22]杨海东,肖宜,王卓民,等.突发性水污染事件溯源方法[J].水科学进展,2014,25(1):122-128.YANG Haidong,XIAO Yi,WANG Zhuomin,et al.On source identification method for sudden water pollution accidents[J].Advances in Water Science,2014,25(1):122-128.
[23]闵涛,周孝德,张世梅,等.对流-扩散方程源项识别反问题的遗传算法[J].水动力学研究与进展,A辑,2004,19(4):520-524.MIN Tao,ZHOU Xiaode,ZHANG Shimei,et al.Genetic algorithm to an inverse problem of source term identification for convection-diffusion equation[J].Journal of Hydrodynamics(Ser.A),2004,19(4):520-524.
[24]牟行洋.基于微分进化算法的污染物源项识别反问题研究[J].水动力学研究与进展(A辑),2011,26(1):24-30.MOU Xingyang.Research on inverse problem of pollution source term identification based on differential evolution algorithm[J].Journal of Hydrodynamics(Ser.A),2011,26(1):24-30.
[25]郑春苗,BENNETT G D.地下水污染物迁移模拟(第二版)[M].北京:高等教育出版社,2009:70-79.