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Kriging-based analytical technique for global sensitivity analysis of systems with multivariate output

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Abstract

Global sensitivity analysis is of great significance for risk assessment of structural systems. In order to efficiently perform sensitivity analysis for systems with multivariate output, this paper adopts Kriging-based analytical (KBA) technique to estimate multivariate sensitivity indices (MSI). Two MSI are studied in this paper, namely, MSI based on principal component analysis (MSI-PCA) and MSI based on covariance decomposition (MSI-CD). For MSI-PCA, Kriging models of inputs and each retained output principal component (PC) are firstly established, and then KBA technique is used to derive the sensitivities associated with each retained PC and the generalized MSI-PCA. For MSI-CD, Kriging model is constructed to map input variables and each time output variable, based on which subset variances and the corresponding MSI-CD are derived by KBA technique. In addition, to avoid constructing Kriging model at each time instant when calculating MSI-CD, a new double-loop Kriging (D-Kriging) method is developed to further improve the efficiency. The accuracy and efficiency of KBA and D-Kriging methods for MSI estimation are tested and discussed by four examples in Sect. 4.

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References

  • Abdi H, Williams LJ (2010) Principal component analysis. Wiley Interdisciplinary Reviews: Computational Statistics 2(4):433–459

    Article  Google Scholar 

  • Aït-Sahalia Y, Xiu D (2019) Principal component analysis of high-frequency data. J Am Stat Assoc 114(525):287–303

    Article  MathSciNet  MATH  Google Scholar 

  • Alexanderian A, Gremaud PA, Smith RC (2020) Variance-based sensitivity analysis for time-dependent processes. Reliab Eng Syst Saf 196:106722

    Article  Google Scholar 

  • Arwade SR, Moradi M, Louhghalam A (2010) Variance decomposition and global sensitivity for structural systems. Eng Struct 32(1):1–10

    Article  Google Scholar 

  • Borgonovo E (2007) A new uncertainty importance measure. Reliab Eng Syst Saf 92(6):771–784

    Article  Google Scholar 

  • Borgonovo E, Plischke E (2016) Sensitivity analysis: a review of recent advances. Eur J Oper Res 248(3):869–887

    Article  MathSciNet  MATH  Google Scholar 

  • Campbell K, McKay MD, Williams BJ (2006) Sensitivity analysis when model outputs are functions. Reliab Eng Syst Saf 91(10–11):1468–1472

    Article  Google Scholar 

  • Chen W, Jin R, Sudjianto A (2005) Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty. J Mech Des 127(5):875–886

    Article  Google Scholar 

  • Cheng K, Lu Z, Zhou Y, Shi Y, Wei Y (2017) Global sensitivity analysis using support vector regression. Appl Math Model 49:587–598

  • Cheng K, Lu Z, Zhang K (2019) Multivariate output global sensitivity analysis using multi-output support vector regression. Struct Multidiscip Optim 59(6):2177–2187

    Article  MathSciNet  Google Scholar 

  • Cheng K, Lu Z, Ling C, Zhou S (2020) Surrogate-assisted global sensitivity analysis: an overview. Struct Multidiscip Optim 61(3):1187–1213

  • Gamboa F, Janon A, Klein T, Lagnoux A (2013) Sensitivity indices for multivariate outputs. CR Math 351(7–8):307–310

  • Garcia-Cabrejo O, Valocchi A (2014) Global sensitivity analysis for multivariate output using polynomial chaos expansion. Reliab Eng Syst Saf 126:25–36

    Article  Google Scholar 

  • Guo M, Hesthaven JS (2019) Data-driven reduced order modeling for time-dependent problems. Comput Methods Appl Mech Eng 345:75–99

    Article  MathSciNet  MATH  Google Scholar 

  • Handbook of uncertainty quantification. New York: Springer, 2017.

  • Kafali C, Grigoriu M (2007) Seismic fragility analysis: Application to simple linear and nonlinear systems. Earthquake Eng Struct Dynam 36(13):1885–1900

    Article  Google Scholar 

  • Lamboni M, Monod H, Makowski D (2011) Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Reliab Eng Syst Saf 96(4):450–459

    Article  Google Scholar 

  • Li L, Lu Z, Chen C (2016a) Moment-independent importance measure of correlated input variable and its state dependent parameter solution. Aerosp Sci Technol 48:281–290

    Article  Google Scholar 

  • Li L, Lu Z, Wu D (2016b) A new kind of sensitivity index for multivariate output. Reliab Eng Syst Saf 147:123–131

    Article  Google Scholar 

  • Ling C, Lu Z, Feng K, Sun B (2019) Efficient numerical simulation methods for estimating fuzzy failure probability based importance measure indices. Struct Multidiscip Optim 59(2):577–593

  • Liu F, Wei P, Tang C, Wang P, Yue Z (2019) Global sensitivity analysis for multivariate outputs based on multiple response Gaussian process model. Reliab Eng Syst Saf 189:287–298

  • Liu Y, Li L, Zhou C, Zhao H (2020) Efficient multivariate sensitivity analysis for dynamic models based on cubature formula. Eng Struct 206:110164

  • Lophaven S N, Nielsen H B, Søndergaard J. DACE: a Matlab Kriging toolbox, version 2.0. Lyngby (Denmark), IMM Technical University of Denmark, 2002.

  • Mara TA, Tarantola S, Annoni P (2015) Non-parametric methods for global sensitivity analysis of model output with dependent inputs. Environ Model Softw 72:173–183

    Article  Google Scholar 

  • Rochman D, Zwermann W, van der Marck SC, Koning AJ, Sjöstrand, Helgesson P, Krzykacz-Hausmann B (2014) Efficient use of Monte Carlo: uncertainty propagation. Nucl Sci Eng 177(3):337–349

    Article  Google Scholar 

  • Saltelli A (2002a) Sensitivity analysis for importance assessment. Risk Anal 22(3):579–590

    Article  Google Scholar 

  • Saltelli A (2002b) Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun 145(2):280–297

    Article  MathSciNet  MATH  Google Scholar 

  • Saltelli A, Marivoet J (1990) Non-parametric statistics in sensitivity analysis for model output: A comparison of selected techniques. Reliab Eng Syst Saf 28(2):229–253

    Article  Google Scholar 

  • Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S (2010) Variance based sensitivity analysis of model output Design and estimator for the total sensitivity index. Comput Phys Commun 181(2):259–270

    Article  MathSciNet  MATH  Google Scholar 

  • Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. John Wiley & Sons

  • Sobola IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1):271–280

    Article  MathSciNet  Google Scholar 

  • Su G, Peng L, Hu L (2017) A Gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis. Struct Saf 68:97–109

    Article  Google Scholar 

  • Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979

    Article  Google Scholar 

  • Sun X, Choi JI (2021) Non-intrusive reduced-order modeling for uncertainty quantification of space–time-dependent parameterized problems. Comput Math Appl 87:50–64

    Article  MathSciNet  MATH  Google Scholar 

  • Sun X, Choi YY, Choi JI (2020) Global sensitivity analysis for multivariate outputs using polynomial chaos-based surrogate models. Appl Math Model 82:867–887

    Article  MathSciNet  MATH  Google Scholar 

  • Wan H, Ren W, Todd MD (2020) Arbitrary polynomial chaos expansion method for uncertainty quantification and global sensitivity analysis in structural dynamics. Mech Syst Signal Process 142:106732

    Article  Google Scholar 

  • Wu Z, Wang D, Okolo P, Hu F, Zhang W (2016) Global sensitivity analysis using a Gaussian radial basis function metamodel. Reliab Eng Syst Saf 154:171–179

  • Xiao S, Lu Z, Qin F (2017) Estimation of the Generalized Sobol’s sensitivity index for multivariate output model using unscented transformation. J Struct Eng 143(5):06016005

    Article  Google Scholar 

  • Xiao S, Lu Z, Wang P (2018a) Multivariate global sensitivity analysis for dynamic models based on energy distance. Struct Multidiscip Optim 57(1):279–291

    Article  MathSciNet  Google Scholar 

  • Xiao S, Lu Z, Wang P (2018b) Global sensitivity analysis based on distance correlation for structural systems with multivariate output. Eng Struct 167:74–83

    Article  Google Scholar 

  • Xiao M, Lu D, Breitkopf P, Raghavan B, Dutta S, Zhang W (2020) On-the-fly model reduction for large-scale structural topology optimization using principal components analysis. Struct Multidisc Optim 62(1):209–230

  • Xu L, Lu Z, Xiao S (2019) Generalized sensitivity indices based on vector projection for multivariate output. Appl Math Model 66:592–610

    Article  Google Scholar 

  • Yun W, Lu Z, Jiang X (2018) An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability. Struct Multidiscip Optim 58(4):1383–1393

    Article  MathSciNet  Google Scholar 

  • Zhang K, Lu Z, Wu D, Zhang Y (2017) Analytical variance based global sensitivity analysis for models with correlated variables. Appl Math Model 45:748–767

  • Zhou S, Lu Z, Ling C, Yun W (2021) Global reliability sensitivity analysis index and its efficient numerical simulation solution in presence of both random and interval hybrid uncertainty. Struct Multidisc Optim 63(2):551–573

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 51875464), and the Fundamental Research Funds for the Central Universities, and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX2022003).

Funding

National Natural Science Foundation of China, Grant No. NSFC 51875464, Luyi Li,Fundamental Research Funds for the Central Universities, Luyi Li, the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University, CX2022003, Yushan Liu.

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Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, P.O. Box 120, Xi’an, Shaanxi Province, 710072, People’s Republic of China

    Yushan Liu, Luyi Li & Zeming Chang

  2. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Pan Wang

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  1. Yushan Liu
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  2. Luyi Li
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  4. Pan Wang
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Correspondence to Luyi Li.

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The authors declared that they have no conflicts of interest in this work, we declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Replication of results

The necessary information for replication of the results has been presented in Sect. 3.2. The relevant codes for the algorithms could be available on request by emailing the first author.

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Liu, Y., Li, L., Chang, Z. et al. Kriging-based analytical technique for global sensitivity analysis of systems with multivariate output. Struct Multidisc Optim 65, 288 (2022). https://doi.org/10.1007/s00158-022-03362-2

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