稀疏偏最小二乘回归-多项式混沌展开代理模型方法
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摘要
为解决传统多项式混沌展开方法在高维全局灵敏度和结构可靠度分析当中存在的维数灾难与多重共线性问题,该文提出一种稀疏偏最小二乘回归-多项式混沌展开代理模型方法。该方法首先采用偏最小二乘回归技术得到多项式混沌展开系数的初步估计,然后根据回归误差阈值允许下的最大稀疏度原则,采用带有惩罚的矩阵分解技术自适应地保留与结构响应相关性强的多项式,并采用偏最小二乘回归得到多项式混沌展开系数的更新估计。通过对展开系数进行简单后处理即可得到Sobol灵敏度指数。在此基础上保留重要输入变量并按新方法重新进行回归可实现对代理模型的精简,从而在不增加计算代价的情况下实现高精度结构可靠度分析。算例结果表明在保证精度的情况下,采用新方法进行全局灵敏度和结构可靠度分析比传统方法在计算效率方面有显著优势。
To circumvent the curse of dimensionality and multicollinearity problems of traditional polynomial chaos expansion approach when analyzing global sensitivity and structural reliability of high-dimensional models, this paper proposes a sparse partial least squares regression-polynomial chaos expansion metamodeling method. Firstly, an initial estimation of polynomial chaos expansion coefficients is obtained with the partial least squares regression. Secondly, according to the principle of maximum sparsity under the allowance of regression error threshold, polynomials which have strong correlation with the structural response are adaptively retained with the penalized matrix decomposition scheme. Next, an updated estimation of the polynomial chaos expansion coefficients is obtained with the partial least squares regression. Sobol sensitivity indices are obtained with a simple post-processing of the expansion coefficients. Finally, the metamodel is greatly simplified by regressing with important inputs, leading to accurate estimations of the failure probability without additional computational cost. The results show that with acceptable accuracies, the new method overperforms the traditional counterpart in terms of computational efficiency when solving high-dimensional global sensitivity and structural reliability analysis problems.
To circumvent the curse of dimensionality and multicollinearity problems of traditional polynomial chaos expansion approach when analyzing global sensitivity and structural reliability of high-dimensional models, this paper proposes a sparse partial least squares regression-polynomial chaos expansion metamodeling method. Firstly, an initial estimation of polynomial chaos expansion coefficients is obtained with the partial least squares regression. Secondly, according to the principle of maximum sparsity under the allowance of regression error threshold, polynomials which have strong correlation with the structural response are adaptively retained with the penalized matrix decomposition scheme. Next, an updated estimation of the polynomial chaos expansion coefficients is obtained with the partial least squares regression. Sobol sensitivity indices are obtained with a simple post-processing of the expansion coefficients. Finally, the metamodel is greatly simplified by regressing with important inputs, leading to accurate estimations of the failure probability without additional computational cost. The results show that with acceptable accuracies, the new method overperforms the traditional counterpart in terms of computational efficiency when solving high-dimensional global sensitivity and structural reliability analysis problems.
引文
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[29]胡军,张树道.基于多项式混沌的全局敏感度分析[J].计算物理,2016(1):1―14.Hu Jun,Zhang Shudao.Global sensitivity analysis based on polynomial chaos[J].Chinese Journal of Computational Physics,2016(1):1―14.(in Chinese)
[30]邬晓敬,张伟伟,宋述芳,等.翼型跨声速气动特性的不确定性及全局灵敏度分析[J].力学学报,2015,47(4):587―595.Wu Xiaojing,Zhang Weiwei,Song Shufang et al.Uncertainty quantification and global sensitivity analysis of transonic aerodynamics about airfoil.Chinese Journal of Theoretical and Applied Mechanics,2015,47(4):587―595.(in Chinese)
[31]宋彦.基于随机响应面法的结构整体可靠度与全局灵敏度分析[D].哈尔滨:哈尔滨工业大学,2016.Song Yan.Global reliability and sensitivity analysis of structures based on stochastic response surface methods[D].Harbin:Harbin Institute of Technology,2016.(in Chinese)
[32]陈光宋,钱林方,吉磊.身管固有频率高效全局灵敏度分析[J].振动与冲击,2015,34(21):31―36.Chen Guangsong,Qian Linfang,Ji Lei.An effective global sensitivity analysis method for natural frequencies of a barrel[J].Journal of Vibration and Shock,2015,34(21):31―36.(in Chinese)
[33]王惠文,吴载斌,孟洁.偏最小二乘回归的线性与非线性方法[M].北京:国防工业出版社,2006.Wang Huiwen,Wu Zaibin,Meng Jie.Linear and nonlinear methods of partial least squares regression[M].Beijing:National Defend Industry Press,2006.(in Chinese)
[34]Deng B,Yun Y,Liang Y,et al.A new strategy to prevent over-fitting in partial least squares models based on model population analysis[J].Analytica Chimica Acta,2015,880:32―41.
[35]Witten D M,Hastie R T T.A penalized matrix decomposition,with applications to sparse principal components and canonical correlation analysis[J].Biostatistics,2009,10(3):515―534.
[36]Pan Q,Dias D.Sliced inverse regression-based sparse polynomial chaos expansions for reliability analysis in high dimensions[J].Reliability Engineering&System Safety,2017,167:484―493.
[37]刘娇,刘敬敏,余波,等.工程结构体系可靠度分析的最新研究进展[J].工程力学,2017,34(增刊1):31―37.Liu Jiao,Liu Jingmin,Yu Bo,et al.Recent research progress on structural system reliability analysis[J].Engineering Mechanics,2017,34(Suppl 1):31―37.(in Chinese)
[38]孙威,黄炎生,杨惠贤.改进响应面法计算钢筋混凝土框架体系可靠度[J].工程力学,2016,33(6):194―201,208.Sun Wei,Huang Yansheng,Yang Huixian.System reliability calculation of RC frame structures by improved response surface method[J].Engineering Mechanics,2016,33(6):194―201,208.(in Chinese)
[2]Breitung K.40 years FORM:Some new aspects?[J].Probabilistic Engineering Mechanics,2015,42:71―77.
[3]Au S K,Beck J L.A new adaptive importance sampling scheme for reliability calculations[J].Structural Safety,1999,21(2):135―158.
[4]Wang Z,Song J.Cross-entropy-based adaptive importance sampling using von Mises-Fisher mixture for high dimensional reliability analysis[J].Structural Safety,2016,59:42―52.
[5]Au S,Beck J L.Estimation of small failure probabilities in high dimensions by subset simulation[J].Probabilistic Engineering Mechanics,2001,16(4):263―277.
[6]Au S.On MCMC algorithm for subset simulation[J].Probabilistic Engineering Mechanics,2016,43:117―120.
[7]Koutsourelakis P S.Reliability of structures in high dimensions.Part II.Theoretical validation[J].Probabilistic Engineering Mechanics,2004,19(4):419―423.
[8]Koutsourelakis P S,Pradlwarter H J,Schueller G I.Reliability of structures in high dimensions,part I:algorithms and applications[J].Probabilistic Engineering Mechanics,2004,19(4):409―417.
[9]Chowdhury R,Rao B N,Prasad A M.High dimensional model representation for structural reliability analysis[J].Communications in Numerical Methods in Engineering,2009,25(4):301―337.
[10]Chakraborty S,Chowdhury R.Assessment of polynomial correlated function expansion for high-fidelity structural reliability analysis[J].Structural Safety,2016,59:9―19.
[11]Bichon B J,Eldred M S,Swiler L P,et al.Efficient global reliability analysis for nonlinear implicit performance functions[J].AIAA journal,2008,46(10):2459―2468.
[12]Hu Z,Mahadevan S.Global sensitivity analysisenhanced surrogate(GSAS)modeling for reliability analysis[J].Structural and Multidisciplinary Optimization,2015,3(53):1―21.
[13]Wiener N.The homogeneous chaos[J].American Journal of Mathematics,1938,60(4):897―936.
[14]Ghanem R,Spanos P D.Polynomial chaos in stochastic finite elements[J].Journal of Applied Mechanics,1990,57(1):197―202.
[15]Choi S K,Grandhi R V,Canfield R A,et al.Polynomial Chaos Expansion with Latin Hypercube Sampling for Estimating Response Variability[J].Aiaa Journal,2004,42(6):1191―1198.
[16]Choi S,Canfield R A,Grandhi R V.Estimation of structural reliability for Gaussian random fields[J].Structures and Infrastructure Engineering,2006,2(3/4):161―173.
[17]李典庆,蒋水华,周创兵.基于非侵入式随机有限元法的地下洞室可靠度分析[J].岩土工程学报,2012,34(1):123―129.Li Dianqing,Jiang Shuihua,Zhou Chuangbing.Reliability analysis of underground rock caverns using non-intrusive stochastic finite element method[J].Chinese Journal of Geotechnical Engineering,2012,34(1):123―129.(in Chinese)
[18]蒋水华,李典庆,周创兵.基于拉丁超立方抽样的边坡可靠度分析非侵入式随机有限元法[J].岩土工程学报,2013,35(增刊2):70―76.Jiang Shuihua,Li Dianqing,Zhou Chuangbing.Non-intrusive stochastic finite element method for slope reliability analysis based on Latin hypercube sampling[J].Chinese Journal of Geotechnical Engineering,2013,35(Suppl 2):70―76.(in Chinese)
[19]蒋水华,冯晓波,李典庆,等.边坡可靠度分析的非侵入式随机有限元法[J].岩土力学,2013,34(8):2347―2354.Jiang Shuihua,Feng Xiaobo,Li Dianqing et al.Reliability analysis of slope using non-intrusive stochastic finite element method[J].Rock and Soil Mechanics,2013,34(8):2347―2354.(in Chinese)
[20]Blatman G,Sudret B.An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis[J].Probabilistic Engineering Mechanics,2010,25(2):183―197.
[21]Blatman G,Sudret B.Adaptive sparse polynomial chaos expansion based on least angle regression[J].Journal of Computational Physics,2011,230(6):2345―2367.
[22]吕震宙,李璐祎,宋述芳,等.不确定性结构系统的重要性分析理论与求解方法[M].北京:科学出版社,2015.LüZhenzhou,Li Luyi,Song Shufang,et al.Analysis theory and computational methods for uncertain structural systems[M].Beijing:Science Press,2015.(in Chinese)
[23]Sobol’,Il’ya Meerovich.On sensitivity estimation for nonlinear mathematical models[J].Matematicheskoe Modelirovanie,1990,2(1):112―118.
[24]Homma T,Saltelli A.Importance measures in global sensitivity analysis of nonlinear models[J].Reliability Engineering&System Safety,1996,52(1):1―17.
[25]Stanfill B,Mielenz H,Clifford D,et al.Simple approach to emulating complex computer models for global sensitivity analysis[J].Environmental Modelling&Software,2015,74(C):140―155.
[26]Oakley J E,O'Hagan A.Probabilistic sensitivity analysis of complex models:a Bayesian approach[J].Journal of the Royal Statistical Society:Series B(Statistical Methodology),2004,66(3):751―769.
[27]Sudret B.Global sensitivity analysis using polynomial chaos expansions[J].Reliability Engineering&System Safety,2008,93(7):964―979.
[28]Blatman G,Sudret B.Efficient computation of global sensitivity indices using sparse polynomial chaos expansions[J].Reliability Engineering&System Safety,2010,95(11):1216―1229.
[29]胡军,张树道.基于多项式混沌的全局敏感度分析[J].计算物理,2016(1):1―14.Hu Jun,Zhang Shudao.Global sensitivity analysis based on polynomial chaos[J].Chinese Journal of Computational Physics,2016(1):1―14.(in Chinese)
[30]邬晓敬,张伟伟,宋述芳,等.翼型跨声速气动特性的不确定性及全局灵敏度分析[J].力学学报,2015,47(4):587―595.Wu Xiaojing,Zhang Weiwei,Song Shufang et al.Uncertainty quantification and global sensitivity analysis of transonic aerodynamics about airfoil.Chinese Journal of Theoretical and Applied Mechanics,2015,47(4):587―595.(in Chinese)
[31]宋彦.基于随机响应面法的结构整体可靠度与全局灵敏度分析[D].哈尔滨:哈尔滨工业大学,2016.Song Yan.Global reliability and sensitivity analysis of structures based on stochastic response surface methods[D].Harbin:Harbin Institute of Technology,2016.(in Chinese)
[32]陈光宋,钱林方,吉磊.身管固有频率高效全局灵敏度分析[J].振动与冲击,2015,34(21):31―36.Chen Guangsong,Qian Linfang,Ji Lei.An effective global sensitivity analysis method for natural frequencies of a barrel[J].Journal of Vibration and Shock,2015,34(21):31―36.(in Chinese)
[33]王惠文,吴载斌,孟洁.偏最小二乘回归的线性与非线性方法[M].北京:国防工业出版社,2006.Wang Huiwen,Wu Zaibin,Meng Jie.Linear and nonlinear methods of partial least squares regression[M].Beijing:National Defend Industry Press,2006.(in Chinese)
[34]Deng B,Yun Y,Liang Y,et al.A new strategy to prevent over-fitting in partial least squares models based on model population analysis[J].Analytica Chimica Acta,2015,880:32―41.
[35]Witten D M,Hastie R T T.A penalized matrix decomposition,with applications to sparse principal components and canonical correlation analysis[J].Biostatistics,2009,10(3):515―534.
[36]Pan Q,Dias D.Sliced inverse regression-based sparse polynomial chaos expansions for reliability analysis in high dimensions[J].Reliability Engineering&System Safety,2017,167:484―493.
[37]刘娇,刘敬敏,余波,等.工程结构体系可靠度分析的最新研究进展[J].工程力学,2017,34(增刊1):31―37.Liu Jiao,Liu Jingmin,Yu Bo,et al.Recent research progress on structural system reliability analysis[J].Engineering Mechanics,2017,34(Suppl 1):31―37.(in Chinese)
[38]孙威,黄炎生,杨惠贤.改进响应面法计算钢筋混凝土框架体系可靠度[J].工程力学,2016,33(6):194―201,208.Sun Wei,Huang Yansheng,Yang Huixian.System reliability calculation of RC frame structures by improved response surface method[J].Engineering Mechanics,2016,33(6):194―201,208.(in Chinese)