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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| Published in | Computer methods in applied mechanics and engineering Vol. 344; pp. 910 - 937 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Amsterdam
Elsevier B.V
01.02.2019
Elsevier BV |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Rahman, Sharif |
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| SubjectTerms | 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 |
| Title | Uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition |
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