Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation

Uncertainty propagation analysis, which assesses the impact of the uncertainty of input variables on responses, is an important component in risk assessment or reliability analysis of structures. This paper proposes an uncertainty propagation analysis method for structures with parameterized probability-box (p-box) representation, which could efficiently compute both the bounds on statistical moments and also the complete probability bounds of the response function. Firstly, based on the sparse grid numerical integration (SGNI) method, an optimized SGNI (OSGNI) is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function (CGF), respectively. Then, using the bounds on the first four cumulants, an optimization procedure based on the saddlepoint approximation is proposed to obtain the whole range of probability bounds of the response function. Through using the saddlepoint approximation, the present approach can achieve a good accuracy in estimating the tail probability bounds of a response function. Finally, two numerical examples and an engineering application are investigated to demonstrate the effectiveness of the proposed method.

[1]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[2]  A. Kiureghian,et al.  Aleatory or epistemic? Does it matter? , 2009 .

[3]  Nancy Reid,et al.  Saddlepoint Methods and Statistical Inference , 1988 .

[4]  David Lamb,et al.  Reliability-Based Design Optimization Using Confidence-Based Model Validation for Insufficient Experimental Data , 2016, Design Automation Conference.

[5]  Gary Tang,et al.  Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation , 2011, Reliab. Eng. Syst. Saf..

[6]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

[7]  Dirk P. Kroese,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[8]  C. Jiang,et al.  A decoupling approach for evidence-theory-based reliability design optimization , 2017 .

[9]  Chao Jiang,et al.  An efficient uncertainty propagation method for parameterized probability boxes , 2016 .

[10]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[11]  Shih-Yu Wang.  General saddlepoint approximations in the bootstrap , 1992 .

[12]  Lin Hu,et al.  An efficient reliability analysis approach for structure based on probability and probability box models , 2017 .

[13]  Byung Man Kwak,et al.  Efficient statistical tolerance analysis for general distributions using three-point information , 2002 .

[14]  N. Wiener.  The Homogeneous Chaos , 1938 .

[15]  Ying Xiong,et al.  A new sparse grid based method for uncertainty propagation , 2010 .

[16]  Jim W. Hall,et al.  Generation, combination and extension of random set approximations to coherent lower and upper probabilities , 2004, Reliab. Eng. Syst. Saf..

[17]  A. Leon-Garcia,et al.  Probability, statistics, and random processes for electrical engineering , 2008 .

[18]  Yang Liu,et al.  Reliability analysis of structures using stochastic response surface method and saddlepoint approximation , 2017 .

[19]  David Gorsich,et al.  Conservative reliability-based design optimization method with insufficient input data , 2016 .

[20]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[21]  Xiaoping Du,et al.  Uncertainty Analysis by Dimension Reduction Integration and Saddlepoint Approximations , 2005, DAC 2005.

[22]  Morgan C. Bruns,et al.  Propagation of Imprecise Probabilities through Black Box Models , 2006 .

[23]  Darrell Whitley,et al.  A genetic algorithm tutorial , 1994, Statistics and Computing.

[24]  Hyeon-Seok Kim,et al.  Validation and updating in a large automotive vibro-acoustic model using a P-box in the frequency domain , 2016 .

[25]  S. Ferson,et al.  Different methods are needed to propagate ignorance and variability , 1996 .

[26]  Daniel Berleant,et al.  Statool: A Tool for Distribution Envelope Determination (DEnv), an Interval-Based Algorithm for Arithmetic on Random Variables , 2003, Reliab. Comput..

[27]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[28]  S. Rahman,et al.  A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics , 2004 .

[29]  Daniel Berleant,et al.  Representation and problem solving with Distribution Envelope Determination (DEnv) , 2004, Reliab. Eng. Syst. Saf..

[30]  C. Paredis,et al.  The Value of Using Imprecise Probabilities in Engineering Design , 2006 .

[31]  S. Huzurbazar.  Practical Saddlepoint Approximations , 1999 .

[32]  R. Mullen,et al.  Interval Monte Carlo methods for structural reliability , 2010 .

[33]  Wei Chen,et al.  A Most Probable Point-Based Method for Efficient Uncertainty Analysis , 2001 .

[34]  Zhonglai Wang,et al.  Reliability sensitivity analysis for structural systems in interval probability form , 2011 .

[35]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[36]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

[37]  C. Jiang,et al.  A new uncertainty propagation method for problems with parameterized probability-boxes , 2018, Reliab. Eng. Syst. Saf..

[38]  Robert L. Mullen,et al.  Structural analysis with probability-boxes , 2012 .

[39]  Zhonglai Wang,et al.  Unified uncertainty analysis by the mean value first order saddlepoint approximation , 2012 .

[40]  Sang-Hoon Lee,et al.  A comparative study of uncertainty propagation methods for black-box-type problems , 2008 .

[41]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[42]  Scott Ferson,et al.  Constructing Probability Boxes and Dempster-Shafer Structures , 2003 .

[43]  H. Daniels.  Saddlepoint Approximations in Statistics , 1954 .

[44]  Christiaan J. J. Paredis,et al.  Numerical Methods for Propagating Imprecise Uncertainty , 2006, DAC 2006.

[45]  S. Rice,et al.  Saddle point approximation for the distribution of the sum of independent random variables , 1980, Advances in Applied Probability.

[46]  Maurice G. Kendall.  The advanced theory of statistics , 1958 .

[47]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[48]  Christiaan J. J. Paredis,et al.  Eliminating Design Alternatives Based on Imprecise Information , 2006 .

[49]  Helen M. Regan,et al.  Equivalence of methods for uncertainty propagation of real-valued random variables , 2004, Int. J. Approx. Reason..

[50]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[51]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[52]  I. Molchanov.  Theory of Random Sets , 2005 .

[53]  Arnold Neumaier.  Clouds, Fuzzy Sets, and Probability Intervals , 2004, Reliab. Comput..

[54]  S. Rahman,et al.  A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics , 2004 .

[55]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[56]  Palle Thoft-Christensen,et al.  Structural Reliability Theory and Its Applications , 1982 .