Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

Arnst, Maarten; Ghanem, Roger; Phipps, Eric Todd; Red-Horse, John

doi:10.1002/nme.4595

Cited by 42 publications

(41 citation statements)

References 30 publications

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“…A systematic construction of PCE of second-order random fields and their use for analyzing the uncertainty propagation in boundary value problems were initiated in [84,85]. Several extensions have been proposed concerning the generalized chaos expansions, the PCE for an arbitrary probability measure, the PCE with random coefficients [12,74,126,162,188,196,217], and the construction of a basis adaptation in homogeneous chaos spaces [213]. Several extensions have been proposed concerning the generalized chaos expansions, the PCE for an arbitrary probability measure, the PCE with random coefficients [12,74,126,162,188,196,217], and the construction of a basis adaptation in homogeneous chaos spaces [213].…”

Section: Brief Overview On the Types Of Stochastic Solversmentioning

confidence: 99%

Uncertainty Quantification

Soize

2017

Interdisciplinary Applied Mathematics

124

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Section: Brief Overview On the Types Of Stochastic Solversmentioning

confidence: 99%

Uncertainty Quantification

Soize

2017

Interdisciplinary Applied Mathematics

124

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“…19 FORM and the likelihood-based decoupling approach cast the feedback-coupled system as a feed-forward system, thus avoiding coupled system analyses and their associated FPI, but at the cost of neglecting the dependence between the inputs and the coupling variables. This can be an effective approach when the sensitivity of the system output to coupling variables is low, but can lead to poor results when sensitivity to coupling variables is high.…”

Section: -15mentioning

confidence: 99%

Multifidelity Uncertainty Propagation in Coupled Multidisciplinary Systems

Chaudhuri

Willcox

2016

18th AIAA Non-Deterministic Approaches Conference

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Fixed point iteration is a common strategy to handle interdisciplinary coupling within a coupled multidisciplinary analysis. For each coupled analysis, this requires a large number of disciplinary high-fidelity simulations to resolve the interactions between different disciplines. When embedded within an uncertainty analysis loop (e.g., with Monte Carlo sampling over uncertain parameters) the number of high-fidelity disciplinary simulations quickly becomes prohibitive, since each sample requires a fixed point iteration and the uncertainty analysis typically involves thousands or even millions of samples. This paper develops a method for uncertainty analysis in feedback-coupled black-box systems that leverages adaptive surrogates to reduce the number of cases for which fixed point iteration is needed. The multifidelity coupled uncertainty propagation method is an iterative process that uses surrogates for approximating the coupling variables and adaptive sampling strategies to refine the surrogates. The adaptive sampling strategies explored in this work are residual error, information gain, and weighted information gain. The surrogate models are adapted in a way that does not compromise accuracy of the uncertainty analysis relative to the original coupled high-fidelity problem.

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“…. , n}, the estimation (η i , η j ) → p H i H j (η i , η j ) of the joint probability density function on R 2 of the R 2 -valued (3,4) and (4,5), graphs of the joint probability density functions (ηi, ηj ) → pH i H j (ηi, ηj) (left figure), and (ηi, ηj ) → pH i H j (ηi, ηj ) for mergo = 10, 000 (right figure).…”

Section: Estimation Of the Probability Density Function Of Hmentioning

confidence: 99%

“…Finally, a numerical application is presented for analyzing the convergence properties. This new class of algorithms for the multimodal case can be useful in the context of uncertainty quantification for direct and inverse problems, and in particular, for the approaches devoted to dimension reduction in chaos expansions for nonlinear coupled problems, when an iterative solver is used (see for instance [2,3,4]). …”

Section: Introductionmentioning

confidence: 99%

Polynomial Chaos Expansion of a Multimodal Random Vector

Soize¹

2015

SIAM/ASA J. Uncertainty Quantification

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Abstract.A methodology and algorithms are proposed for constructing the polynomial chaos expansion (PCE) of multimodal random vectors. An algorithm is developed for generating independent realizations of any multimodal multivariate probability measure that is constructed from a set of independent realizations using the Gaussian kernel-density estimation method. The PCE is then performed with respect to this multimodal probability measure, for which the realizations of the polynomial chaos are computed with an adapted algorithm. Finally, a numerical application is presented for analyzing the convergence properties.Key words. multimodal probability distribution, polynomial chaos, random vector, high dimension AMS subject classifications. 62H10, 62H12, 62H20, 60E10, 60E05, 60G35, 60G601. Introduction. In 1991, R. Ghanem [16] has proposed (1) an efficient construction of the polynomial chaos expansion (PCE) [8] for representing second-order stochastic processes and random fields, and (2) to use it for solving boundary value problems with uncertain parameters by a spectral approach and the stochastic finite elements. Since 1991, numerous works have been published in the area of the PCE and of its use in the spectral approaches for solving linear and nonlinear stochastic boundary value problems, and some associated statistical inverse problems (see for instance [1,9,10,11,13,17,18,27,31,34,42,44]). Several extensions have been proposed concerning generalized chaos expansions, the PCE for an arbitrary probability measure, the PCE with random coefficients [14,28,38,40,49,50], and recently, the construction of a basis adaptation in homogeneous chaos spaces [48]. Although several works have been devoted to the acceleration of stochastic convergence of the PCE (see for instance [19,24,29,48]), the question relative to the speed of convergence (which can be very low) of the PCE for a multimodal probability distribution on R n has been little addressed. Recently, a procedure through mixtures of PCE has been proposed in [33] for the onedimension case. In this paper, we propose a methodology for the PCE of a multimodal R m -valued random variable. This problem belongs to the class of the PCE with respect to an arbitrary probability measure. The framework of the developments presented in the paper is motivated by the difficulty encountered for the PCE of a random vector for which its probability density function is multimodal, and for which it is known that the speed of convergence of the PCE can be low. Nevertheless, the method proposed is very general and goes beyond multimodality. In the context of the statistical inverse problem related to the identification of a PCE of a random vector, one does not know if the unknown probability density function is unimodal or is multimodal. So the method proposed allows for accelerating the speed of convergence in all the cases. We propose an algorithm for generating independent realizations of the multimodal probability measure on R n , which is constructed from a set of realizations us...

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Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

Cited by 42 publications

References 30 publications

Uncertainty Quantification

Uncertainty Quantification

Multifidelity Uncertainty Propagation in Coupled Multidisciplinary Systems

Polynomial Chaos Expansion of a Multimodal Random Vector

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