Spectral Methods for Uncertainty Quantification

Maître, Olivier Le; Knio, Omar M.

doi:10.1007/978-90-481-3520-2

Cited by 825 publications

(552 citation statements)

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“…Two relevant polynomial approximation strategies that can be conveniently applied to the problem at hand are the Stochastic Galerkin [1,[7][8][9][10] and the Stochastic Collocation methods [2,[11][12][13], which are a projection technique and an interpolation technique, respectively. In this work, we reconsider the quasi-optimal Stochastic Galerkin method proposed in the previous work [3], and provide rigorous convergence results in the special case in which the analyticity region contains a polydisc in the complex plane C N .…”

Section: Introductionmentioning

confidence: 99%

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Beck

Nobile

Tamellini

et al. 2014

Computers & Mathematics with Applications

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a b s t r a c tIn this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane C N . We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates.

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Section: Introductionmentioning

confidence: 99%

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Beck

Nobile

Tamellini

et al. 2014

Computers & Mathematics with Applications

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“…It is not seldom to meet in practise deterministic models with a number of unknowns N of order 10 5 . According to (14) and Table 1, the unknown number NxP can be quickly very huge (of order 10 8 ) if the number K of random input parameters is higher than a dozen.…”

Section: Approximation Methodsmentioning

confidence: 99%

“…Under separability condition on the behaviour law, this system of equations can be written in the form of (21) which alleviates the storage space requirement. Dedicated solvers can be applied [1,14] but it does not decrease the size of the equation system. Model Order Reduction Methods like Proper Orthogonal Decomposition (POD), Reduced Basis Method enables to reduce the stochastic problem (9) to solve to an order R ≤≤ N (N is the number of DoF's of the spatial mesh) [49].…”

Section: Intrusive Methodsmentioning

confidence: 99%

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Clénet

2016

Scientific Computing in Electrical Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. To cite this version :Stephane CLÉNET -Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics -2016Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.eu Approximation Methods to solve Stochastic Problems in Computational ElectromagneticsStéphane ClénetAbstract To account for uncertainties on model parameters, the stochastic approach can be used. The model parameters as well as the outputs are then random fields or variables. Several methods are available in the literature to solve stochastic models like sampling methods, perturbation methods or approximation methods. In this paper, we propose an overview on the solution of stochastic problems in computational electromagnetics using approximation methods. Some applications will be presented in order to illustrate the possibilities offered by the approximation methods but also their current limitations due to the curse of dimensionality. Finally, recent numerical techniques proposed in the literature to face the curse of dimensionality are presented for non-intrusive and intrusive approaches.

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“…These last two decades, a growing interest has been devoted to spectral stochastic methods, which provide an explicit representation of the random model output as a function of the basic random parameters modeling the input uncertainties [63,64,97,98,99]. An approximation of the random model output is sought on suitable functional approximation bases.…”

Section: Propagation Of Uncertainties or What Are The Methods To Solvmentioning

confidence: 99%

“…(ii.1) -The first one corresponds to the spectral methods such as the Polynomial Chaos representations (see [63,64] and also [65,66,67,68,69,70,71,72,73]) which can be applied in infinite dimension for stochastic processes and random fields, which allow the effective construction of mapping h to be carried out and which allow any random variable X in L 2 N , to be written as…”

Section: Types Of Representation For the Stochastic Modeling Of Uncermentioning

confidence: 99%

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

Soize

2013

Journal of Sound and Vibration

140

106

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To cite this version:Christian Soize. Stochastic modeling of uncertainties in computational structural dynamics -Recent theoretical advances. Journal of Sound and Vibration, Elsevier, 2013, 332 (10) AbstractThis paper deals with a short overview on stochastic modeling of uncertainties. We introduce the types of uncertainties, the variability of real systems, the types of probabilistic approaches, the representations for the stochastic models of uncertainties, the construction of the stochastic models using the maximum entropy principle, the propagation of uncertainties, the methods to solve the stochastic dynamical equations, the identification of the prior and the posterior stochastic models, the robust updating of the computational models and the robust design with uncertain computational models. We present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties, and for their identification with experimental data. We also present the construction of the posterior stochastic model of uncertainties using the Bayesian method when experimental data are available.

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Spectral Methods for Uncertainty Quantification

Cited by 825 publications

References 0 publications

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

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