Practical uncertainty quantification analysis involving statistically dependent random variables

Lee, Dongjin; Rahman, Sharif

doi:10.1016/j.apm.2020.03.041

Cited by 15 publications

(5 citation statements)

References 21 publications

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“…For an arbitrary PDF f X (x), the matrix G S,m cannot be determined exactly, yet it can be accurately estimated with numerical integration and/or sampling methods [19].…”

Section: Measure-consistent Orthonormal Polynomialsmentioning

confidence: 99%

“…However, in practice, input variables are often correlated or dependent. Indeed, neglecting the correlation in input random variables, whether emanating from loads, material properties, or manufacturing variables may produce inaccurate or unknown risky designs [27,19].…”

Section: Introductionmentioning

confidence: 99%

“…While standard MCS is can handle dependent random variables directly, it requires high-fidelity solves and therefore can be too expensive computationally. A practical version of the GPCE was recently introduced to effectively solve UQ and design optimization problems under arbitrary, dependent input random variables [19,20,21]. This work makes it possible to obtain the multivariate orthonormal polynomial basis consistent with any non-product-type probability measure of input numerically, instead of an analytical expression by a Rodrigues-type formula used in the prequel [34].…”

Section: Introductionmentioning

confidence: 99%

“…19: end procedure 20: Construct the mth-order Fourier polynomial expansion y H, m(x) of high-fidelity output y H (x) (28). 21: By replacing y with y H, m, construct y S,m (19) and then perform Algorithm 1 to estimate VaR β and CVaR β .…”

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confidence: 99%

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Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Kramer¹

2022

Preprint

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Digital twin models allow us to continuously assess the possible risk of damage and failure of a complex system. Yet high-fidelity digital twin models can be computationally expensive, making quick-turnaround assessment challenging. Towards this goal, this article proposes a novel bi-fidelity method for estimating the conditional value-at-risk (CVaR) for nonlinear systems subject to dependent and high-dimensional inputs. For models that can be evaluated fast, a method that integrates the dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) approximation with a standard sampling-based CVaR estimation is proposed. For expensive-to-evaluate models, a new bi-fidelity method is proposed that couples the DD-GPCE with a Fourier-polynomial expansions of the mapping between the stochastic low-fidelity and high-fidelity output data to ensure computational efficiency. The method employs a measure-consistent orthonormal polynomial in the random variable of the low-fidelity output to approximate the high-fidelity output. Numerical results for a structural mechanics truss with 36-dimensional (dependent random variable) inputs indicate that the DD-GPCE method provides very accurate CVaR estimates that require much lower computational effort than standard GPCE approximations. A second example considers the realistic problem of estimating the risk of damage to a fiber-reinforced composite laminate. The high-fidelity model is a finite element simulation that is prohibitively expensive for risk analysis, such as CVaR computation. Here, the novel bi-fidelity method can accurately estimate CVaR as it includes low-fidelity models in the estimation procedure and uses only a few high-fidelity model evaluations to significantly increase accuracy.

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“…For an arbitrary PDF f X (x), the matrix G S,m cannot be determined exactly, yet it can be accurately estimated with numerical integration and/or sampling methods [19].…”

Section: Measure-consistent Orthonormal Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Kramer¹

2022

Preprint

Self Cite

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“…For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ variables [25], [26], [27], [28], dominating measures [29], [30], and the construction of orthogonal basis functions for correlated random variables [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. However, these methods often require information about the probability distribution.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Method to Quantify Correlated Uncertainties

Jung

Choi

2023

IEEE Access

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Polynomial chaos (PC) has been proven to be an efficient method for uncertainty quantification, but its applicability is limited by two strong assumptions: the mutual independence of random variables and the requirement of exact knowledge about the distribution of the random variables. We describe a new data-driven method for dealing with correlated multivariate random variables for uncertainty quantification that requires only observed data of the random variables. It is based on the transformation of correlated random variables into independent random variables. We use singular value decomposition as a transformation strategy that does not require information about the probability distribution. For the transformed random variables, we can construct the PC basis to build a surrogate model. This approach provides an additional benefit of quantifying high-dimensional uncertainties by combining our method with the analysisof-variance (ANOVA) method. We demonstrate in several numerical examples that our proposed approach leads to accurate solutions with a much smaller number of simulations compared to the Monte Carlo method.INDEX TERMS Correlated random variables, high dimension, polynomial chaos expansion, uncertainty quantification.

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Robust design optimization under dependent random variables by a generalized polynomial chaos expansion

Lee

Rahman

2021

Struct Multidisc Optim

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Practical uncertainty quantification analysis involving statistically dependent random variables

Cited by 15 publications

References 21 publications

Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Data-Driven Method to Quantify Correlated Uncertainties

Robust design optimization under dependent random variables by a generalized polynomial chaos expansion

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