Uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition

This paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables. The analysis focuses on a new, generalized version of polynomial dimensional decomposition (PDD), referred to as GPDD, entailing hierarchically ordered measure-consistent...

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Bibliographic Details
Published inComputer methods in applied mechanics and engineering Vol. 344; pp. 910 - 937
Main Author Rahman, Sharif
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 01.02.2019
Elsevier BV
Subjects
ANOVA
Boundary value problems
Complex systems
Decomposition
Dependence
Dependent variables
Eigenvalues
Error analysis
Formulas (mathematics)
Independent variables
Linear equations
Multivariate orthogonal polynomials
Non-product-type probability measures
Polynomials
Random variables
Statistical analysis
Thermal expansion
Uncertainty analysis
Variance
Multivariate orthogonal polynomials
ANOVA
Non-product-type probability measures
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Summary:This paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables. The analysis focuses on a new, generalized version of polynomial dimensional decomposition (PDD), referred to as GPDD, entailing hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent variables. Under a few prescribed assumptions, GPDD exists for any square-integrable output random variable and converges in mean-square to the correct limit. New analytical formulae are proposed to calculate the mean and variance of a GPDD approximation of a general output variable in terms of the expansion coefficients and second-moment properties of multivariate orthogonal polynomials. However, unlike in PDD, calculating the coefficients of GPDD requires solving a coupled system of linear equations. Besides, the variance formula of GPDD contains extra terms due to statistical dependence among input variables. The extra terms disappear when the input variables are statistically independent, reverting GPDD to PDD. Two numerical examples, the one derived from a stochastic boundary-value problem and the other entailing a random eigenvalue problem, illustrate second-moment error analysis and estimation of the probabilistic characteristics of eigensolutions.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2018.09.026