Uncertainty propagation analysis by an extended sparse grid technique

Abstract

In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.

Keywords

uncertainty propagation analysis / extended sparse grid / maximum entropy principle / extended Gauss integration / Rosenblatt transformation / high-order moments analysis

References

[1] Lee S H, Chen W. A comparative study of uncertainty propagation methods for black-box-type problems. Structural and Multidisciplinary Optimization, 2009, 37(3): 239–253

[2] Wang X, Wang L, Qiu Z. Response analysis based on smallest interval-set of parameters for structures with uncertainty. Applied Mathematics and Mechanics, 2012, 33(9): 1153–1166

[3] Wang X, Wang L, Qiu Z. A feasible implementation procedure for interval analysis method from measurement data. Applied Mathematical Modelling, 2014, 38(9–10): 2377–2397

[4] Qiu Z P, Wang L. The need for introduction of non-probabilistic interval conceptions into structural analysis and design. Science China. Physics, Mechanics & Astronomy, 2016, 59(11): 114632

[5] Gu X, Renaud J E, Batill S M, Worst case propagated uncertainty of multidisciplinary systems in robust design optimization. Structural and Multidisciplinary Optimization, 2000, 20(3): 190–213

[6] Li M, Azarm S. Multiobjective collaborative robust optimization with interval uncertainty and interdisciplinary uncertainty propagation. Journal of Mechanical Design, 2008, 130(8): 081402

[7] Li G, Zhang K. A combined reliability analysis approach with dimension reduction method and maximum entropy method. Structural and Multidisciplinary Optimization, 2011, 43(1): 121–134

[8] Jiang Z, Li W, Apley D W, A spatial-random-process based multidisciplinary system uncertainty propagation approach with model uncertainty. Journal of Mechanical Design, 2015, 137(10): 101402

[9] Mazo J, El Badry A T, Carreras J, Uncertainty propagation and sensitivity analysis of thermo-physical properties of phase change materials (PCM) in the energy demand calculations of a test cell with passive latent thermal storage. Applied Thermal Engineering, 2015, 90: 596–608

[10] Li M, Mahadevan S, Missoum S, Special issue: Simulation-based design under uncertainty. Journal of Mechanical Design, 2016, 138(11): 110301

[11] Madsen H O, Krenk S, Lind N C. Methods of structural safety. Mineola: Dover Publications, 2006

[12] Wilson B M, Smith B L. Taylor-series and Monte-Carlo-method uncertainty estimation of the width of a probability distribution based on varying bias and random error. Measurement Science & Technology, 2013, 24(3): 035301

[13] Rochman D, Zwermann W, van der Marck S C, Efficient use of Monte Carlo: Uncertainty propagation. Nuclear Science and Engineering, 2014, 177(3): 337–349

[14] Hong J, Shaked S, Rosenbaum R K, Analytical uncertainty propagation in life cycle inventory and impact assessment: Application to an automobile front panel. International Journal of Life Cycle Assessment, 2010, 15(5): 499–510

[15] Xu L. A proportional differential control method for a time-delay system using the Taylor expansion approximation. Applied Mathematics and Computation, 2014, 236: 391–399

[16] Low B. FORM, SORM, and spatial modeling in geotechnical engineering. Structural Safety, 2014, 49: 56–64

[17] Lim J, Lee B, Lee I. Post optimization for accurate and efficient reliability-based design optimization using second-order reliability method based on importance sampling and its stochastic sensitivity analysis. International Journal for Numerical Methods in Engineering, 2015, 107(2): 93–108

[18] Lee I, Choi K K, Gorsich D. Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO). International Journal for Numerical Methods in Engineering, 2010, 82(1): 26–46

[19] Sudret B. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 2008, 93(7): 964–979

[20] Kersaudy P, Sudret B, Varsier N, A new surrogate modeling technique combining Kriging and polynomial chaos expansions—Application to uncertainty analysis in computational dosimetry. Journal of Computational Physics, 2015, 286: 103–117

[21] Rajabi M M, Ataie-Ashtiani B, Simmons C T. Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations. Journal of Hydrology (Amsterdam), 2015, 520: 101–122

[22] Rahman S, Xu H. A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilistic Engineering Mechanics, 2004, 19(4): 393–408

[23] Xu H, Rahman S. A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. International Journal for Numerical Methods in Engineering, 2004, 61(12): 1992–2019

[24] Nobile F, Tempone R, Webster C G. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 2008, 46(5): 2309–2345

[25] Xiong F, Greene S, Chen W, A new sparse grid based method for uncertainty propagation. Structural and Multidisciplinary Optimization, 2010, 41(3): 335–349

[26] He J, Gao S, Gong J. A sparse grid stochastic collocation method for structural reliability analysis. Structural Safety, 2014, 51: 29–34

[27] Smolyak S A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk SSSR, 1963, 4: 240–243

[28] Novak E, Ritter K. High dimensional integration of smooth functions over cubes. Numerische Mathematik, 1996, 75(1): 79–97

[29] Novak E, Ritter K. Simple cubature formulas with high polynomial exactness. Constructive Approximation, 1999, 15(4): 499–522

[30] Bathe K, Wilson E. Stability and accuracy analysis of direct integration methods. Earthquake Engineering & Structural Dynamics, 1972, 1(3): 283–291

[31] Tao J, Zeng X, Cai W, Stochastic sparse-grid collocation algorithm (SSCA) for periodic steady-state analysis of nonlinear system with process variations. In: Proceedings of the 2007 Asia and South Pacific Design Automation Conference. IEEE, 2007, 474–479

[32] Jia B, Xin M, Cheng Y. Sparse Gauss-Hermite quadrature filter for spacecraft attitude estimation. In: Proceedings of the 2010 American Control Conference. Baltimore: IEEE, 2010, 2873–2878

[33] Petvipusit K R, Elsheikh A H, Laforce T C, Robust optimisation of CO₂ sequestration strategies under geological uncertainty using adaptive sparse grid surrogates. Computational Geosciences, 2014, 18(5): 763–778

[34] Chen H, Cheng X, Dai C, Accuracy, efficiency and stability analysis of sparse-grid quadrature Kalman filter in near space hypersonic vehicles. In: Proceedings of Position, Location and Navigation Symposium-PLANS 2014, 2014 IEEE/ION. Monterey: IEEE, 2014, 27–36

[35] KendallM G, Stuart A. The Advanced Theory of Statistics Volume 1: Distribution Theory. London: Charles Griffin & Company, 1958

[36] Press W H, Teukolsky S A, Vetterling W T, Numerical recipes in C. Cambridge: Cambridge University Press, 1996

[37] Ghosh D D, Olewnik A. Computationally efficient imprecise uncertainty propagation. Journal of Mechanical Design, 2013, 135(5): 051002

[38] Ahlfeld R, Belkouchi B, Montomoli F. SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos. Journal of Computational Physics, 2016, 320: 1–16

[39] Patterson T. Modified optimal quadrature extensions. Numerische Mathematik, 1993, 64(1): 511–520

[40] Kronrod A S.Nodes and Weights of Quadrature Formulas: Sixteen-place Tables. New York: Consultants Bureau, 1965

[41] Patterson T. The optimum addition of points to quadrature formulae. Mathematics of Computation, 1968, 22(104): 847–856

[42] Genz A, Keister B D. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics, 1996, 71(2): 299–309

[43] Scarth C, Cooper J E, Weaver P M, Uncertainty quantification of aeroelastic stability of composite plate wings using lamination parameters. Composite Structures, 2014, 116: 84–93

[44] Feinberg J, Langtangen H P. Chaospy: An open source tool for designing methods of uncertainty quantification. Journal of Computational Science, 2015, 11: 46–57

[45] Huang B, Du X. Uncertainty analysis by dimension reduction integration and saddlepoint approximations. Journal of Mechanical Design, 2006, 128(1): 26–33

[46] Gerstner T, Griebel M. Numerical integration using sparse grids. Numerical Algorithms, 1998, 18(3–4): 209–232

[47] Heiss F, Winschel V. Likelihood approximation by numerical integration on sparse grids. Journal of Econometrics, 2008, 144(1): 62–80

[48] Jaynes E T. Information theory and statistical mechanics. Physical Review, 1957, 106(4): 620–630

[49] Phillips S J, Anderson R P, Schapire R E. Maximum entropy modeling of species geographic distributions. Ecological Modelling, 2006, 190(3–4): 231–259

[50] Mohammad-Djafari A. A Matlab program to calculate the maximum entropy distributions. In: Smith C R, Erickson G J, Neudorfer P O, eds. Maximum Entropy and Bayesian Methods. Dordrecht: Springer, 1992, 221–233

[51] Yeo S K, Chun J H, Kwon Y S. A 3-D X-band T/R module package with an anodized aluminum multilayer substrate for phased array radar applications. IEEE Transactions on Advanced Packaging, 2010, 33(4): 883–891

[52] Pamies Porras M J, Bertuch T, Loecker C, An AESA antenna comprising an RF feeding network with strongly coupled antenna ports. IEEE Transactions on Antennas and Propagation, 2015, 63(1): 182–194