Statistical Moments of Polynomial Dimensional Decomposition

Rahman, Sharif

doi:10.1061/(asce)em.1943-7889.0000117

Cited by 28 publications

(26 citation statements)

References 8 publications

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“…The fundamental impediment to practical computability is frequently related to the high dimension of the multivariate integration or interpolation problem, known as the curse of dimensionality. The recently invented polynomial dimensional decomposition (PDD) method alleviates the curse of dimensionality to some extent by splitting a high‐dimensional output function into a finite sum of simpler component functions that are arranged with respect to the degree of interaction among input random variables. In addition, the method exploits the smoothness properties of a stochastic response, whenever possible, by expanding its component functions in terms of measure‐consistent orthogonal polynomials, leading to closed‐form expressions of the second‐moment characteristics of a stochastic solution.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

Yadav

Rahman

2013

Numerical Meth Engineering

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SUMMARYThe central theme of this paper is multiplicative polynomial dimensional decomposition (PDD) methods for solving high‐dimensional stochastic problems. When a stochastic response is dominantly of multiplicative nature, the standard PDD approximation, predicated on additive function decomposition, may not provide sufficiently accurate probabilistic solutions of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function, a broad range of orthonormal polynomial bases for Fourier‐polynomial expansions of component functions, and a dimension‐reduction or sampling technique for estimating the expansion coefficients. Three numerical problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative PDD is more efficient or accurate than the additive PDD. The results show that indeed, both the factorized and logarithmic PDD approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the new methods to solve industrial‐scale problems. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

Yadav

Rahman

2013

Numerical Meth Engineering

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“…of the S-variate, mth-order A-PDD approximation matches the exact mean E½yðXÞ, regardless of S or m, and the approximate variance [11] σ 2 S;m ≔E½ðỹ S;m ðXÞÀE½ỹ S;m ðXÞÞ 2…”

Section: Additive Pddmentioning

confidence: 80%

“…It is elementary to show that the approximate variance in Eq. (12) approaches the exact variance of y when S-N and m-1 [11]. The mean-square convergence ofỹ S;m is guaranteed as y, and its component functions are all members of the associated Hilbert spaces.…”

Section: Additive Pddmentioning

confidence: 98%

“…Applying the expectation operator onỹ S;m ðXÞ and ðỹ S;m ðXÞÀy ∅ Þ 2 and noting Propositions 1 and 2, the mean [11] E½ỹ S;m ðXÞ ¼ y ∅…”

Section: Additive Pddmentioning

confidence: 99%

“…Unlike Eqs. (11) and (12), the mean and variance ofŷ S;m ðXÞ, respectively defined as do not produce closed-form or analytic expressions in terms of y ∅ and C uj juj if S is selected arbitrarily. This is a drawback of F-PDD when compared with A-PDD.…”

Section: Factorized Pddmentioning

confidence: 99%

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A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

Yadav

Rahman

2014

Probabilistic Engineering Mechanics

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Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

Liu

Zhao

2022

Earthq Engng Struct Dyn

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Based on polynomial dimensional decomposition (PDD), a novel method for analysing nonstationary random vibration of structures with uncertain parameters is proposed in this paper. The uncertain structural parameters are assumed to be independent random variables, with certain types of probability distributions; thus, the nonstationary random vibration responses of the structures become stochastic functions of these uncertain parameters. A dimensional decomposition and Fourier expansion were applied to calculate the statistical characteristics of the nonstationary random vibration responses of the structures. To calculate the conditional power spectral density (CPSD) and conditional standard deviation (CSD) of the nonstationary random vibration responses, CPSD and CSD were explicitly expressed using expansion coefficients and polynomial basis. Subsequently, the probability distributions of the structural responses can be effectively calculated in terms of these explicit expressions. Dimension-reduction integration and Gauss numerical integration were applied to calculate expansion coefficients. The validity of the proposed method was verified by numerical examples, including a single-degree-of-freedom system and a multi-degree-offreedom system (reinforced concrete shear wall). The results indicate that the proposed method is as accurate as Monte Carlo simulation (MCS), whereas the proposed method requires considerably smaller number of nonstationary random vibration analyses than MCS. Moreover, the nonstationary random vibration analysis of a long-span cable-stayed bridge demonstrates that the proposed method is feasible for quantifying the uncertainties of the nonstationary random vibration responses of complex engineering structures with uncertain parameters.

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Statistical Moments of Polynomial Dimensional Decomposition

Cited by 28 publications

References 8 publications

Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

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