Efficient multivariate sensitivity analysis for dynamic models based on cubature formula

Abstract Variance-based sensitivity analysis (SA) has been frequently applied to dynamic models with multivariate output. The generalized sensitivity indices are defined by combining principal components analysis with analysis of variance to synthesize the influence of each input on the whole dynamic output. In order to efficiently perform global SA on dynamic models, two efficient algorithms based on cubature formula are proposed in this paper to estimate these generalized variance-based sensitivity indices. The new algorithms are double-loop nested cubature formula (DLCF) and single loop cubature formula (SLCF). The DLCF method estimates the sensitivity indices by decreasing the dimensionality of the input variables procedurally, while SLCF method performs SA through extending the dimensionality of the inputs. Both of them can make full use of the advantages of the cubature formula, and provide efficient estimates for the generalized variance-based sensitivity indices. The numerical and engineering examples demonstrate that both the proposed algorithms can avoid the expensive computational cost associated with the sampling methods, and improve the SA of the dynamic models significantly.

[1]  David L. Darmofal,et al.  Higher-Dimensional Integration with Gaussian Weight for Applications in Probabilistic Design , 2005, SIAM J. Sci. Comput..

[2]  David Makowski,et al.  Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models , 2011, Reliab. Eng. Syst. Saf..

[3]  Fabrice Gamboa,et al.  Sensitivity analysis for multidimensional and functional outputs , 2013, 1311.1797.

[4]  Bruno Sudret,et al.  Distribution-based global sensitivity analysis using polynomial chaos expansions , 2010 .

[5]  Jorge Cadima,et al.  Principal component analysis: a review and recent developments , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Paul Geladi,et al.  Principal Component Analysis , 1987, Comprehensive Chemometrics.

[7]  Zdeněk Kala,et al.  Global sensitivity analysis of reliability of structural bridge system , 2019, Engineering Structures.

[8]  Yongbo Peng,et al.  Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes , 2017 .

[9]  Zhen-zhou Lü,et al.  Moment-independent importance measure of basic random variable and its probability density evolution solution , 2010 .

[10]  Jianbing Chen,et al.  Probability density evolution analysis of engineering structures via cubature points , 2012 .

[11]  Andrea Saltelli,et al.  An effective screening design for sensitivity analysis of large models , 2007, Environ. Model. Softw..

[12]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[13]  Qiao Liu,et al.  A new computational method of a moment-independent uncertainty importance measure , 2009, Reliab. Eng. Syst. Saf..

[14]  Zhenzhou Lu,et al.  AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability , 2020 .

[15]  Hermann G. Matthies,et al.  Non-probabilistic interval process model and method for uncertainty analysis of transient heat transfer problem , 2019, International Journal of Thermal Sciences.

[16]  Carlos Alberto Conceição António,et al.  Reliability-based design optimization and uncertainty quantification for optimal conditions of composite structures with non-linear behavior , 2017 .

[17]  George M. Phillips,et al.  A survey of one-dimensional and multidimensional numerical integration , 1980 .

[18]  Thierry A. Mara,et al.  Bayesian sparse polynomial chaos expansion for global sensitivity analysis , 2017 .

[19]  Wei Wang,et al.  Nonlinear system stochastic response determination via fractional equivalent linearization and Karhunen-Loève expansion , 2017, Commun. Nonlinear Sci. Numer. Simul..

[20]  Z. Qiu,et al.  Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters , 2017 .

[21]  Andrea Saltelli,et al.  Sensitivity Analysis for Importance Assessment , 2002, Risk analysis : an official publication of the Society for Risk Analysis.

[22]  Jianbing Chen,et al.  Improving Point Selection in Cubature by a New Discrepancy , 2013, SIAM J. Sci. Comput..

[23]  Kaichao Zhang,et al.  An efficient sampling method for variance-based sensitivity analysis , 2017 .

[24]  Zhenzhou Lu,et al.  Time-dependent failure credibility analysis and its optimization based computational methods , 2019 .

[25]  Emanuele Borgonovo,et al.  A new uncertainty importance measure , 2007, Reliab. Eng. Syst. Saf..

[26]  A. H. Stroud,et al.  Some fifth degree integration formulas for symmetric regions , 1966 .

[27]  Brian J. Williams,et al.  Sensitivity analysis when model outputs are functions , 2006, Reliab. Eng. Syst. Saf..

[28]  Albert J. Valocchi,et al.  Global Sensitivity Analysis for multivariate output using Polynomial Chaos Expansion , 2014, Reliab. Eng. Syst. Saf..

[29]  Paola Annoni,et al.  Non-parametric methods for global sensitivity analysis of model output with dependent inputs , 2015, Environ. Model. Softw..

[30]  Zhenzhou Lu,et al.  Variance-based sensitivity analysis for models with correlated inputs and its state dependent parameter solution , 2017, Structural and Multidisciplinary Optimization.

[31]  Chong Wang,et al.  Novel model calibration method via non-probabilistic interval characterization and Bayesian theory , 2019, Reliab. Eng. Syst. Saf..

[32]  A. Stroud,et al.  Approximate integration formulas for certain spherically symmetric regions , 1963 .

[33]  Arghavan Louhghalam,et al.  Variance decomposition and global sensitivity for structural systems , 2010 .

[34]  J. McNamee,et al.  Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas , 1967 .

[35]  Nicolas Victoir.  Asymmetric Cubature Formulae with Few Points in High Dimension for Symmetric Measures , 2004, SIAM J. Numer. Anal..

[36]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[37]  Xiaoping Du,et al.  Reliability analysis for hydrokinetic turbine blades , 2012 .

[38]  Louis J. Durlofsky,et al.  Regularized kernel PCA for the efficient parameterization of complex geological models , 2016, J. Comput. Phys..

[39]  Zhao-Hui Lu,et al.  Evaluation of Moments of Performance Functions Based on Efficient Cubature Formulation , 2017 .

[40]  Christoph Schwab,et al.  Karhunen-Loève approximation of random fields by generalized fast multipole methods , 2006, J. Comput. Phys..

[41]  Juan Manuel Górriz,et al.  Early diagnosis of Alzheimer's disease based on partial least squares, principal component analysis and support vector machine using segmented MRI images , 2015, Neurocomputing.

[42]  I. Jolliffe.  Principal Component Analysis , 2002 .

[43]  Zhenzhou Lu,et al.  Global sensitivity analysis based on distance correlation for structural systems with multivariate output , 2018, Engineering Structures.

[44]  Zhenzhou Lu,et al.  Efficient numerical algorithm of profust reliability analysis: An application to wing box structure , 2018, Aerospace Science and Technology.