阻尼最小二乘算法研究综述
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摘要
最小二乘法(LSM)是一种经典的数学优化方法,基本原理是通过最小化误差的平方和以确定数据与方程之间的最佳函数匹配关系。LSM简单易行,在许多领域应用广泛。众多学者对LSM进行广泛研究,提出了各种改进方法,其中比较著名的是阻尼最小二乘法(LMA)。通过研读近几年有关阻尼最小二乘算法文献,将不同学者对阻尼最小二乘算法的改进策略研究进行梳理。
As a classical mathematical optimization method,least square method(LSM)determines the optimal function matching relationship between data and function by minimizing the sum of squares of errors.LSM is simple and easy to use,and has been widely used in many fields.Many scholars have conducted extensive research on LSM,and proposed various improvement methods,among which the most famous is Levenberg Marquardt Algorithm(LMA).By studying the literatures about LMA published in recent years,this paper summarizes and analyzes the improvement strategies of LMA.
As a classical mathematical optimization method,least square method(LSM)determines the optimal function matching relationship between data and function by minimizing the sum of squares of errors.LSM is simple and easy to use,and has been widely used in many fields.Many scholars have conducted extensive research on LSM,and proposed various improvement methods,among which the most famous is Levenberg Marquardt Algorithm(LMA).By studying the literatures about LMA published in recent years,this paper summarizes and analyzes the improvement strategies of LMA.
引文
[1]STIGLER S M.The history of statistics:the measurement of uncertainty before 1900[M].Cambridge:Belknap Press of Harvard University Press,1986.
[2]Legendre A M.Nouvelles méthodes pour la détermination des orbites des comètes[M].Paris:Nabu Press,1805.
[3]LEVENBERG K.A method for the solution of certain non-linear problems in least squares[J].Quarterly of Applied Mathematics,1944,2(4):164-168.
[4]WYNNE C G.Lens designing by electronic digital computer:I[J].Proceedings of the Physical Society,1959,73(5):777-787.
[5]MORRISON D D.Methods for nonlinear least squares problems and convergence proofs[J].Proceedings of the Seminar on Tracking Programs and Orbit Determination,1960,2012(1):1409-1415.
[6]王大麒,朱匀华,陈志恬.阻尼最小二乘法中权的选择原理[J].中山大学学报,1980(1):24-39.
[7]朱匀华,陈志恬.阻尼最小二乘法的目标函数分离形式(Ⅰ)[J].中山大学学报,1983(1):65-71.
[8]朱匀华,陈志恬.阻尼最小二乘法的目标函数分离形式(Ⅱ)[J].中山大学学报,1984(2):72-74.
[9]李以柏,郑永梅.阻尼最小二乘法的一种改进[J].厦门大学学报:自然科学版,1985,24(4):522-526.
[10]陈钟琦.高阻尼因子对阻尼最小二乘法效果的影响和克服[J].现代地质,1988,2(2):255-265.
[11]卢进,王小华,郭姝言,等.基于递推阻尼最小二乘法的电力系统频率跟踪[J].电子科技,2014,27(12):17-19.
[12]GAVIN H P.The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems[EB/OL].http://people.duke.edu/~hpgavin/ce281/lm.pdf.
[13]LOURAKIS M I A.A brief description of the Levenberg-Marquardt algorithm implemented by levmar[EB/OL].http://users.ics.forth.gr/lourakis/levmar/levmar.pdf.
[14]MADSEN K,NIELSEN N B,TINGLEFF O.Methods for nonlinear least squares problems[J].Society for Industrial&Applied Mathematics,2004(1):1409-1415.
[15]MARQUARDT D W.An algorithm for least-squares estimation of nonlinear parameters[J].Journal of the Society for Industrial and Applied Mathematics,1963,11(2):431-441.
[16]NIELSEN H B.Damping parameter in marquardt′s method[EB/OL].http://www.imm.dtu.dk/documents/ftp/tr99/tr05_99.pdf.
[17]PRESS W H,TEUKOSKY S A,VETTERLING W T,et al.Numerical recipes in C[M].United Kingdom:Cambridge University Press,1992.
[18]TRANSTRUM M K,SETHNA J P.Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization[EB/OL].https://arxiv.org/abs/1201.5885.
[19]MARK K T,BENJAMIN B M,JAMES P S.Why are nonlinear fits to data so challenging[J].Physical Review Letters,2010,104(6):1-4.
[20]陈德豪,张琰.常用阻尼最小二乘算法的改进[J].物测科技,1994(1):31-52.
[21]孙若昧,郑斯华,马林,等.阻尼最小二乘法联合测定震源位置和介质速度参数[J].地震,1986(4):29-37.
[22]何宗海.用阻尼最小二乘法研究云南地区的Q值分布[J].西北大学学报:自然科学版,1995,25(3):237-240.
[23]宋林平.阻尼加权最小二乘地震走时层析成象[J].计算物理,1995,12(4):499-504.
[24]KALACHAND,戴成泰.运用阻尼最小二乘法的广角地震反射时间反演[J].石油物探译丛,1995(5):27-37.
[25]阮百尧,葛为中.奇异值分解法与阻尼最小二乘法的对比[J].物探化探计算技术,1997,19(1):47-49.
[26]赵龙,刘淮,陈哲.递推阻尼最小二乘及其在INS/双星组合中的应用[J].北京航空航天大学学报,2003,29(12):1136-1138.
[27]王进,李欣然,苏盛.用阻尼最小二乘优化算法实现电力系统综合静态负荷建模[J].长沙电力学院学报:自然科学版,2004,19(1):40-46.
[28]孙明玮,张奇,邵继法,等.鲁棒递推阻尼最小二乘算法[J].航空计算技术,2003,33(1):22-25.
[29]张国芹,朱长青,王光霞.GIS数据误差处理的阻尼最小二乘算法[J].测绘学院学报,2004,21(1):8-10.
[30]马英杰,虎胆·吐马尔拜,沈冰.利用阻尼最小二乘法求解Van Genuchten方程参数[J].农业工程学报,2005,21(8):179-180.
[31]JOSE P.The solution of nonlinear inverse problems and the Levenberg-Marquardt method[J].Geophysics,2007,72(4):1-16.
[32]孙秩超.瞬变电磁数据改进阻尼最小二乘拟合算法研究[D].长春:吉林大学,2009.
[33]李培.基于奇异值分解的瑞雷面波加权阻尼最小二乘反演[D].长沙:中南大学,2011.
[34]张皓,刘建松,黄金辉,等.最优化阻尼最小二乘法重力二维反演方法及其应用[J].中国西部科技,2011,11(31):34-36.
[35]王晟,胡志云,张振荣,等.阻尼最小二乘算法CARS光谱温度拟合[J].强激光与粒子束,2012,12(11):2565-2570.
[36]张亚兵,陈同俊,崔若飞,等.基于阻尼最小二乘法的煤层裂隙P波方位属性预测[J].煤田地质与勘探,2012,40(4):79-81,85.
[37]王亚强.改进的阻尼最小二乘层析算法及影响因素分析[D].秦皇岛:燕山大学,2013.
[38]王园园,刘斌,王晨.基于阻尼最小二乘法的瞬变电磁反演算法研究[J].电子测试,2013(1):1-7.
[39]王寅,赵南京,刘文清,等.阻尼最小二乘法用于激光诱导击穿光谱重叠特征谱线分离提取[J].光谱学与光谱分析,2015,35(2):309-314.
[40]陈诚.三轴天线座阻尼最小二乘跟踪策略研究[J].现代雷达,2015,37(7):44-47.
[41]平晶晶.基于阻尼最小二乘法的表面等离激元定向耦合器的优化[D].南京:南京航空航天大学,2016.
[42]陈恒,严良俊,代小威,等.模拟退火法与阻尼最小二乘法联合反演岩石激电谱参数[J].工程地球物理学报,2016,13(2):170-174.
[43]项伟,白征东,汤晓禹.阻尼最小二乘法在任意欧拉角坐标转换中的应用[J].大地测量与地球动力学,2016,36(2):167-170.
[44]林茂琼,陈增强,袁著祉.基于阻尼最小二乘法的鲁棒自校正预测控制器[J].南开大学学报:自然科学版,1998,31(2):79-83.
[45]林茂琼,陈增强,贺江峰,等.基于阻尼最小二乘法的神经网络自校正一步预测控制器[J].控制与决策,1999,14(2):165-168.
[46]王阿霞,张文波.阻尼最小二乘法在射线追踪中的应用[J].西安工程学院学报,2001,23(1):43-45.
[2]Legendre A M.Nouvelles méthodes pour la détermination des orbites des comètes[M].Paris:Nabu Press,1805.
[3]LEVENBERG K.A method for the solution of certain non-linear problems in least squares[J].Quarterly of Applied Mathematics,1944,2(4):164-168.
[4]WYNNE C G.Lens designing by electronic digital computer:I[J].Proceedings of the Physical Society,1959,73(5):777-787.
[5]MORRISON D D.Methods for nonlinear least squares problems and convergence proofs[J].Proceedings of the Seminar on Tracking Programs and Orbit Determination,1960,2012(1):1409-1415.
[6]王大麒,朱匀华,陈志恬.阻尼最小二乘法中权的选择原理[J].中山大学学报,1980(1):24-39.
[7]朱匀华,陈志恬.阻尼最小二乘法的目标函数分离形式(Ⅰ)[J].中山大学学报,1983(1):65-71.
[8]朱匀华,陈志恬.阻尼最小二乘法的目标函数分离形式(Ⅱ)[J].中山大学学报,1984(2):72-74.
[9]李以柏,郑永梅.阻尼最小二乘法的一种改进[J].厦门大学学报:自然科学版,1985,24(4):522-526.
[10]陈钟琦.高阻尼因子对阻尼最小二乘法效果的影响和克服[J].现代地质,1988,2(2):255-265.
[11]卢进,王小华,郭姝言,等.基于递推阻尼最小二乘法的电力系统频率跟踪[J].电子科技,2014,27(12):17-19.
[12]GAVIN H P.The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems[EB/OL].http://people.duke.edu/~hpgavin/ce281/lm.pdf.
[13]LOURAKIS M I A.A brief description of the Levenberg-Marquardt algorithm implemented by levmar[EB/OL].http://users.ics.forth.gr/lourakis/levmar/levmar.pdf.
[14]MADSEN K,NIELSEN N B,TINGLEFF O.Methods for nonlinear least squares problems[J].Society for Industrial&Applied Mathematics,2004(1):1409-1415.
[15]MARQUARDT D W.An algorithm for least-squares estimation of nonlinear parameters[J].Journal of the Society for Industrial and Applied Mathematics,1963,11(2):431-441.
[16]NIELSEN H B.Damping parameter in marquardt′s method[EB/OL].http://www.imm.dtu.dk/documents/ftp/tr99/tr05_99.pdf.
[17]PRESS W H,TEUKOSKY S A,VETTERLING W T,et al.Numerical recipes in C[M].United Kingdom:Cambridge University Press,1992.
[18]TRANSTRUM M K,SETHNA J P.Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization[EB/OL].https://arxiv.org/abs/1201.5885.
[19]MARK K T,BENJAMIN B M,JAMES P S.Why are nonlinear fits to data so challenging[J].Physical Review Letters,2010,104(6):1-4.
[20]陈德豪,张琰.常用阻尼最小二乘算法的改进[J].物测科技,1994(1):31-52.
[21]孙若昧,郑斯华,马林,等.阻尼最小二乘法联合测定震源位置和介质速度参数[J].地震,1986(4):29-37.
[22]何宗海.用阻尼最小二乘法研究云南地区的Q值分布[J].西北大学学报:自然科学版,1995,25(3):237-240.
[23]宋林平.阻尼加权最小二乘地震走时层析成象[J].计算物理,1995,12(4):499-504.
[24]KALACHAND,戴成泰.运用阻尼最小二乘法的广角地震反射时间反演[J].石油物探译丛,1995(5):27-37.
[25]阮百尧,葛为中.奇异值分解法与阻尼最小二乘法的对比[J].物探化探计算技术,1997,19(1):47-49.
[26]赵龙,刘淮,陈哲.递推阻尼最小二乘及其在INS/双星组合中的应用[J].北京航空航天大学学报,2003,29(12):1136-1138.
[27]王进,李欣然,苏盛.用阻尼最小二乘优化算法实现电力系统综合静态负荷建模[J].长沙电力学院学报:自然科学版,2004,19(1):40-46.
[28]孙明玮,张奇,邵继法,等.鲁棒递推阻尼最小二乘算法[J].航空计算技术,2003,33(1):22-25.
[29]张国芹,朱长青,王光霞.GIS数据误差处理的阻尼最小二乘算法[J].测绘学院学报,2004,21(1):8-10.
[30]马英杰,虎胆·吐马尔拜,沈冰.利用阻尼最小二乘法求解Van Genuchten方程参数[J].农业工程学报,2005,21(8):179-180.
[31]JOSE P.The solution of nonlinear inverse problems and the Levenberg-Marquardt method[J].Geophysics,2007,72(4):1-16.
[32]孙秩超.瞬变电磁数据改进阻尼最小二乘拟合算法研究[D].长春:吉林大学,2009.
[33]李培.基于奇异值分解的瑞雷面波加权阻尼最小二乘反演[D].长沙:中南大学,2011.
[34]张皓,刘建松,黄金辉,等.最优化阻尼最小二乘法重力二维反演方法及其应用[J].中国西部科技,2011,11(31):34-36.
[35]王晟,胡志云,张振荣,等.阻尼最小二乘算法CARS光谱温度拟合[J].强激光与粒子束,2012,12(11):2565-2570.
[36]张亚兵,陈同俊,崔若飞,等.基于阻尼最小二乘法的煤层裂隙P波方位属性预测[J].煤田地质与勘探,2012,40(4):79-81,85.
[37]王亚强.改进的阻尼最小二乘层析算法及影响因素分析[D].秦皇岛:燕山大学,2013.
[38]王园园,刘斌,王晨.基于阻尼最小二乘法的瞬变电磁反演算法研究[J].电子测试,2013(1):1-7.
[39]王寅,赵南京,刘文清,等.阻尼最小二乘法用于激光诱导击穿光谱重叠特征谱线分离提取[J].光谱学与光谱分析,2015,35(2):309-314.
[40]陈诚.三轴天线座阻尼最小二乘跟踪策略研究[J].现代雷达,2015,37(7):44-47.
[41]平晶晶.基于阻尼最小二乘法的表面等离激元定向耦合器的优化[D].南京:南京航空航天大学,2016.
[42]陈恒,严良俊,代小威,等.模拟退火法与阻尼最小二乘法联合反演岩石激电谱参数[J].工程地球物理学报,2016,13(2):170-174.
[43]项伟,白征东,汤晓禹.阻尼最小二乘法在任意欧拉角坐标转换中的应用[J].大地测量与地球动力学,2016,36(2):167-170.
[44]林茂琼,陈增强,袁著祉.基于阻尼最小二乘法的鲁棒自校正预测控制器[J].南开大学学报:自然科学版,1998,31(2):79-83.
[45]林茂琼,陈增强,贺江峰,等.基于阻尼最小二乘法的神经网络自校正一步预测控制器[J].控制与决策,1999,14(2):165-168.
[46]王阿霞,张文波.阻尼最小二乘法在射线追踪中的应用[J].西安工程学院学报,2001,23(1):43-45.
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