Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems

Arnst, Maarten; Soize, Christian; Ghanem, Roger

doi:10.1137/120894403

Cited by 20 publications

(14 citation statements)

References 16 publications

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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%

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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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Section: Estimation Of the Probability Density Function Of Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial Chaos Expansion of a Multimodal Random Vector

Soize¹

2015

SIAM/ASA J. Uncertainty Quantification

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“…The target input density, q .x I m /, only needs to be evaluated at the proposal samples as specified in Algorithm 1. A procedure for evaluating this density at the proposal samples, using Monte Carlo simulation to evaluate Equation (18) and Lemma 2 to construct q .x J m jx K m /, is given in Algorithm 2. Algorithm 2 avoids the challenge of estimating high-dimensional densities with kernel density methods by assembling the large densities q .x J m / and q .x K m / using the smaller dimensional component densities q x V i and q x T i .…”

Section: Lemmamentioning

confidence: 99%

“…Multiple models coupled together through a handful of scalars, which are represented using truncated Karhunen-Loève expansions, have been studied for multiphysics systems [17]. A hybrid method that combines Monte Carlo sampling and spectral methods for solving stochastic coupled problems has also been proposed [18,19]. The hybrid approach partitions the coupled problem into subsidiary subproblems, which use Monte Carlo sampling methods if the subproblem depends on a very large number of uncertain parameters and spectral methods if the subproblem depends on only a small or moderate number of uncertain parameters.…”

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confidence: 99%

A decomposition‐based approach to uncertainty analysis of feed‐forward multicomponent systems

Amaral

Allaire

Willcox

2014

Numerical Meth Engineering

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SUMMARYTo support effective decision making, engineers should comprehend and manage various uncertainties throughout the design process. Unfortunately, in today's modern systems, uncertainty analysis can become cumbersome and computationally intractable for one individual or group to manage. This is particularly true for systems comprised of a large number of components. In many cases, these components may be developed by different groups and even run on different computational platforms. This paper proposes an approach for decomposing the uncertainty analysis task among the various components comprising a feed-forward system and synthesizing the local uncertainty analyses into a system uncertainty analysis. Our proposed decomposition-based multicomponent uncertainty analysis approach is shown to be provably convergent in distribution under certain conditions. The proposed method is illustrated on quantification of uncertainty for a multidisciplinary gas turbine system and is compared to a traditional system-level Monte Carlo uncertainty analysis approach. Copyright © 2014 John Wiley & Sons, Ltd.

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“…11 The real system is made up of two plates connected together through a complex joint constituted of two smaller plates tightened by 2 lines of 20 bolts. Plate 1 (left) and plate 2 (right) are isotropic thin plates.…”

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confidence: 99%