Locally Refined Adaptive Sparse Surrogate-Based Approach for Uncertainty Quantification
Publication: Journal of Engineering Mechanics
Volume 145, Issue 5
Abstract
Two novel surrogate-based approaches have been developed for uncertainty quantification of engineering systems. In doing so, two well-known techniques, namely, high dimensional model representation (HDMR) and Kriging, have been integrated. Specifically, the trend portion of Kriging has been replaced by HDMR such that the approximation accuracy may be enhanced. The improvement in accuracy is the result of the fact that the proposed hybrid surrogate model performs a two-tier approximation, first capturing the global variation in the functional space using a set of component functions by HDMR and subsequently interpolating the local fluctuations by Kriging. Additionally, to improve the computational cost of this proposed model, feature selection approaches, namely, least absolute shrinkage and selection operator and least angle regression have been employed. These efficient schemes utilized to determine the relevant unknown coefficients induces adaptive sparsity for the proposed surrogate models. The performance of the proposed approaches has been assessed by solving an analytical and practical engineering problem. The results illustrate excellent performance of the proposed approaches in terms of both approximation accuracy and computational effort.
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References
Alis, O. F., and H. Rabitz. 2001. “Efficient implementation of high dimensional model representations.” J. Math. Chem. 29 (2): 127–142. https://doi.org/10.1023/A:1010979129659.
Basudhar, A., and S. Missoum. 2008. “Adaptive explicit decision functions for probabilistic design and optimization using support vector machines.” Comput. Struct. 86 (19–20): 1904–1917. https://doi.org/10.1016/j.compstruc.2008.02.008.
Blatman, G., and B. Sudret. 2010. “An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis.” Probab. Eng. Mech. 25 (2): 183–197. https://doi.org/10.1016/j.probengmech.2009.10.003.
Blatman, G., and B. Sudret. 2011. “Adaptive sparse polynomial chaos expansion based on least angle regression.” J. Comput. Phys. 230 (6): 2345–2367. https://doi.org/10.1016/j.jcp.2010.12.021.
Chastaing, G., F. Gamboa, and C. Prieur. 2012. “Generalized Hoeffding-Sobol decomposition for dependent variables—Application to sensitivity analysis.” Electr. J. Stat. 6: 2420–2448. https://doi.org/10.1214/12-EJS749.
Chatterjee, T., S. Chakraborty, and R. Chowdhury. 2016. “A bi-level approximation tool for the computation of FRFs in stochastic dynamic systems.” Mech. Syst. Signal Process. 70–71 (Mar): 484–505. https://doi.org/10.1016/j.ymssp.2015.09.001.
Chatterjee, T., S. Chakraborty, and R. Chowdhury. 2019. “A critical review of surrogate assisted robust design optimization.” Arch. Comput. Methods Eng. 26 (1): 245–274. https://doi.org/10.1007/s11831-017-9240-5.
Chatterjee, T., and R. Chowdhury. 2017. “Improved sparse approximation models for stochastic computations.” In Handbook of neural computation, edited by V. E. B. Pijush Samui and S. Sekhar Roy. London: Elsevier.
Cortes, C., and V. Vapnik. 1995. “Support-vector networks.” Mach. Learn. 20 (3): 273–297. https://doi.org/10.1007/BF00994018.
Dai, H., H. Zhang, and W. Wang. 2015. “A multiwavelet neural network-based response surface method for structural reliability analysis.” Comput.-Aided Civ. Infrastruct. Eng. 30 (2): 151–162. https://doi.org/10.1111/mice.12086.
Deng, J. 2006. “Structural reliability analysis for implicit performance function using radial basis function network.” Int. J. Solids Struct. 43 (11–12): 3255–3291. https://doi.org/10.1016/j.ijsolstr.2005.05.055.
Doostan, A., and G. Iaccarino. 2009. “A least-squares approximation of partial differential equations with high-dimensional random inputs.” J. Comput. Phys. 228 (12): 4332–4345. https://doi.org/10.1016/j.jcp.2009.03.006.
Doostan, A., A. Validi, and G. Iaccarino. 2013. “Non-intrusive low-rank separated approximation of high-dimensional stochastic models.” Comput. Methods Appl. Mech. Eng. 263 (Aug): 42–55. https://doi.org/10.1016/j.cma.2013.04.003.
Dubourg, V. 2011. “Adaptive surrogate models for reliability analysis and reliability-based design optimization.” Ph.D. thesis, Laboratoire de Mécanique et Ingénieries, Universite Blaise Pascal.
Dubourg, V., B. Sudret, and M. Bourinet. 2011. “Reliability-based design optimization using Kriging surrogates and subset simulation.” Struct. Multi. Opt. 44 (5): 673–690. https://doi.org/10.1007/s00158-011-0653-8.
Echard, B., N. Gayton, and M. Lemaire. 2011. “AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation.” Struct. Saf. 33 (2): 145–154. https://doi.org/10.1016/j.strusafe.2011.01.002.
Efron, B., T. Hastie, I. Johnstone, and R. Tibshirani. 2004. “Least angle regression.” Ann. Stat. 32 (2): 407–499. https://doi.org/10.1214/009053604000000067.
Elsheikh, A. H., I. Hoteit, and M. F. Wheeler. 2014. “Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates.” Comput. Methods Appl. Mech. Eng. 269 (Feb): 515–537. https://doi.org/10.1016/j.cma.2013.11.001.
Friedman, J. H. 1991. “Multivariate adaptive regression splines.” Ann. Stat. 19 (1): 1–67. https://doi.org/10.1214/aos/1176347963.
Gautschi, W. 1996. “Orthogonal polynomials: Applications and computation.” Acta Numerica 5: 45–119. https://doi.org/10.1017/S0962492900002622.
Goswami, S., S. Ghosh, and S. Chakraborty. 2016. “Reliability analysis of structures by iterative improved response surface method.” Struct. Saf. 60 (May): 56–66. https://doi.org/10.1016/j.strusafe.2016.02.002.
Grandvalet, Y. 1998. “Least absolute shrinkage is equivalent to quadratic penalization.” In Proc., 8th Int. Conf. on Artificial Neural Networks, 201–206. London: Springer.
Jacquelin, E., S. Adhikari, J. Sinou, and M. I. Friswell. 2015. “Polynomial chaos expansion and steady-state response of a class of random dynamical systems.” J. Eng. Mech. 141 (4): 04014145. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000856.
Jekabsons, G. 2016. “ARESLab: Adaptive regression splines toolbox for Matlab/Octave.” Accessed May 15, 2016. https://www.cs.rtu.lv/jekabsons/regression.html.
Jiang, L., and X. Li. 2015. “Multi-element least square HDMR methods and their applications for stochastic multiscale model reduction.” J. Comput. Phys. 294 (Aug): 439–461. https://doi.org/10.1016/j.jcp.2015.03.066.
Jin, R., W. Chen, and T. Simpson. 2001. “Comparative studies of metamodeling techniques under multiple modeling criteria.” Struct. Multi. Optim. 23 (1): 1–13. https://doi.org/10.1007/s00158-001-0160-4.
Kai, B., R. Li, and H. Zou. 2010. “Local composite quantile regression smoothing: An efficient and safe alternative to local polynomial regression.” J. R. Stat. Soc. Ser. B-Stat. Methodol. 72 (1): 49–69. https://doi.org/10.1111/j.1467-9868.2009.00725.x.
Kang, S.-C., H.-M. Koh, and J. F. Choo. 2010. “An efficient response surface method using moving least squares approximation for structural reliability analysis.” Probab. Eng. Mech. 25 (4): 365–371. https://doi.org/10.1016/j.probengmech.2010.04.002.
Kaymaz, I. 2005. “Application of Kriging method to structural reliability problems.” Struct. Saf. 27 (2): 133–151. https://doi.org/10.1016/j.strusafe.2004.09.001.
Kersaudy, P., B. Sudret, N. Varsier, O. Picon, and J. Wiart. 2015. “A new surrogate modeling technique combining Kriging and polynomial chaos expansions—Application to uncertainty analysis in computational dosimetry.” J. Comput. Phys. 286 (Apr): 103–117. https://doi.org/10.1016/j.jcp.2015.01.034.
Kim, S.-H., and S.-W. Na. 1997. “Response surface method using vector projected sampling points.” Struct. Saf. 19 (1): 3–19. https://doi.org/10.1016/S0167-4730(96)00037-9.
Kwok, J. 2000. “The evidence framework applied to support vector machines.” IEEE Trans. Neural Networks 11 (5): 1162–1173. https://doi.org/10.1109/72.870047.
Li, E., H. Wang, and G. Li. 2012. “High dimensional model representation (HDMR) coupled intelligent sampling strategy for nonlinear problems.” Comput. Phys. Commun. 183 (9): 1947–1955. https://doi.org/10.1016/j.cpc.2012.04.017.
Li, G., and H. Rabitz. 2008. “Regularized random- sampling high dimensional model representation (RS-HDMR).” J. Math. Chem. 43 (3): 1207–1232. https://doi.org/10.1007/s10910-007-9250-x.
Li, G., and H. Rabitz. 2012. “General formulation of HDMR component functions with independent and correlated variables.” J. Math. Chem. 50 (1): 99–130. https://doi.org/10.1007/s10910-011-9898-0.
Li, G., C. Rosenthal, and H. Rabitz. 2001. “High dimensional model representation.” J. Phys. Chem. A 105 (33): 7765–7777. https://doi.org/10.1021/jp010450t.
Liu, C., S. X. Yang, and L. Deng. 2015. “A comparative study for least angle regression on NIR spectra analysis to determine internal qualities of navel oranges.” Expert Syst. Appl. 42 (22): 8497–8503. https://doi.org/10.1016/j.eswa.2015.07.005.
Lophaven, S., H. Nielson, and J. Sondergaard. 2002. DACE A MATLAB kriging toolbox. IMM-TR-2002-12. Kongens Lyngby, Technical Univ. of Denmark.
Ma, X., and N. Zabaras. 2010. “An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations.” J. Comput. Phys. 229 (10): 3884–3915. https://doi.org/10.1016/j.jcp.2010.01.033.
Mareš, T., E. Janouchová, and A. Kučerová. 2016. “Artificial neural networks in the calibration of nonlinear mechanical models.” Adv. Eng. Softw. 95 (May): 68–81. https://doi.org/10.1016/j.advengsoft.2016.01.017.
Marquardt, D. 1963. “An algorithm for least squares estimation of nonlinear parameters.” SIAM J. Appl. Math. 11 (2): 431–441. https://doi.org/10.1137/0111030.
McKay, M. D., R. J. Beckman, and W. J. Conover. 1979. “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.” Technometrics 21 (2): 239–245.
Mukhopadhyay, T., S. Chakraborty, S. Dey, S. Adhikari, and R. Chowdhury. 2018. “A critical assessment of kriging model variants for high-fidelity uncertainty quantification in dynamics of composite shells.” Arch. Comput. Methods Eng. 24 (3): 495–518. https://doi.org/10.1007/s11831-016-9178-z.
Osborne, M. R., B. Presnell, and B. Turlach. 2000. “A new approach to variable selection in least squares problems.” IMA J. Numer. Anal. 20 (3): 389–403. https://doi.org/10.1093/imanum/20.3.389.
Osborne, M. R., and B. A. Turlach. 2011. “A homotopy algorithm for the quantile regression lasso and related piecewise linear problems.” J. Comput. Graph. Stat. 20 (4): 972–987. https://doi.org/10.1198/jcgs.2011.09184.
Patelli, E., M. Broggi, M. Angelis, and M. Beer. 2014. “OpenCossan: An efficient open tool for dealing with epistemic and aleatory uncertainties.” In Vulnerability, uncertainty, and risk, 2564–2573. Reston, VA: ASCE.
Rabitz, H., and Ö. F. Aliş. 1999. “General foundations of high dimensional model representations.” J. Math. Chem. 25 (2–3): 197–233. https://doi.org/10.1023/A:1019188517934.
Rasmussen, C. E., and C. K. I. Williams. 2006. Gaussian processes for machine learning. Cambridge, MA: MIT Press.
Sacks, J., W. Welch, T. Mitchell, and H. Wynn. 1989. “Design and analysis of computer experiments.” Stat. Sci. 4 (4): 409–423. https://doi.org/10.1214/ss/1177012413.
Sjöstrand, K., L. Clemmensen, R. Larsen, and B. Ersboll. 2018. “Spasm: A MATLAB toolbox for sparse statistical modeling.” J. Stat. Softw. 84 (10): 1–37. https://doi.org/10.18637/jss.v084.i10.
Sobol, I., and Y. Levitan. 1999. “On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index.” Comput. Phys. Commun. 117 (1–2): 52–61. https://doi.org/10.1016/S0010-4655(98)00156-8.
Tibshirani, R. 1996. “Regression shrinkage and selection via the Lasso.” J. R. Stat. Soc. B 58 (1): 267–288.
Tunga, M. A., and M. Demiralp. 2006. “Hybrid high dimensional model representation (HHDMR) on the partitioned data.” J. Comput. Appl. Math. 185 (1): 107–132. https://doi.org/10.1016/j.cam.2005.01.030.
Validi, A. 2014. “Low-rank separated representation surrogates of high-dimensional stochastic functions: Application in Bayesian inference.” J. Comput. Phys. 260 (Mar): 37–53. https://doi.org/10.1016/j.jcp.2013.12.024.
Volpi, S., M. Diez, N. J. Gaul, H. Song, U. Iemma, K. K. Choi, E. F. Campana, and F. Stern. 2015. “Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification.” Struct. Multi. Optim. 51 (2): 347–368. https://doi.org/10.1007/s00158-014-1128-5.
Wang, H., L. Tang, and G. Y. Li. 2011. “Adaptive MLS-HDMR metamodeling techniques for high dimensional problems.” Expert Syst. Appl. 38 (11): 14117–14126. https://doi.org/10.1016/j.eswa.2011.04.220.
Xiu, D., and G. E. Karniadakis. 2002a. “Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos.” Comput. Methods Appl. Mech. Eng. 191 (43): 4927–4948. https://doi.org/10.1016/S0045-7825(02)00421-8.
Xiu, D., and G. E. Karniadakis. 2002b. “The Wiener–Askey polynomial Chaos for stochastic differential equations.” SIAM J. Sci. Comput. 24 (2): 619–644. https://doi.org/10.1137/S1064827501387826.
Xiu, D., and G. E. Karniadakis. 2003. “Modeling uncertainty in flow simulations via generalized polynomial chaos.” J. Comput. Phys. 187 (1): 137–167. https://doi.org/10.1016/S0021-9991(03)00092-5.
Xu, H., and S. Rahman. 2005. “Decomposition methods for structural reliability analysis.” Prob. Eng. Mech. 20 (3): 239–250. https://doi.org/10.1016/j.probengmech.2005.05.005.
Yuan, Q., and D. Liang. 2011. “A new multiple subdomain RS-HDMR method and its application to tropospheric alkane photochemistry model.” Numer. Anal. Model. Ser. B 2 (1): 73–90.
Yun, W., Z. Lu, X. Jiang, and L. Zhang. 2018. “Borgonovo moment independent global sensitivity analysis by Gaussian radial basis function meta-model.” Appl. Math. Model. 54 (Feb): 378–392. https://doi.org/10.1016/j.apm.2017.09.048.
Zhao, L., K. Choi, and I. Lee. 2010. “A metamodeling method using dynamic kriging and sequential sampling.” In Proc., 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conf. Fort Worth, TX: American Institute of Aeronautics and Astronautics.
Zio, E. 2006. “A study of the bootstrap method for estimating the accuracy of artificial neural networks in predicting nuclear transient processes.” IEEE Trans. Nucl. Sci. 53 (3): 1460–1478. https://doi.org/10.1109/TNS.2006.871662.
Zou, H., and T. Hastie. 2005. “Regularization and variable selection via the elastic net.” J. R. Stat. Soc.: Ser. B 67 (2): 301–320. https://doi.org/10.1111/j.1467-9868.2005.00503.x.
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Published In
Journal of Engineering Mechanics
Volume 145 • Issue 5 • May 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jun 20, 2017
Accepted: Oct 30, 2018
Published online: Feb 27, 2019
Published in print: May 1, 2019
Discussion open until: Jul 27, 2019
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