On the convergence of generalized polynomial chaos expansions

Ernst, Oliver G.; Mugler, Antje; Starkloff, Hans-Jörg; Ullmann, Elisabeth

doi:10.1051/m2an/2011045

Cited by 362 publications

(323 citation statements)

References 45 publications

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“…[22]. As a word of caution, we can note that the work [23] showed that there exist probability measures, such as the lognormal one, which admit a family of orthonormal polynomials that however does not form a basis for L 2 ϱ (Γ ), i.e. there exist functions in L 2 ϱ (Γ ) that cannot be approximated with arbitrary precision by linear combinations of such orthonormal polynomials.…”

Section: Galerkin Polynomial Approximation In the Stochastic Dimensionmentioning

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: Galerkin Polynomial Approximation In the Stochastic Dimensionmentioning

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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“…This proposal is based on the following key fact: the weighting functions associated to the so-called Wiener-Askey scheme, that correspond to classical families of orthogonal polynomials, are identical to the probability density functions of some standard random variables. Recently, in [10] authors provide conditions under which gPC expansions converge in the mean-square sense to the correct limit, which constitutes an extension of paper of Cameron and Martin [7] where mean-square convergence of homogeneous chaos was established.…”

Section: Introductionmentioning

confidence: 92%

Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach

Chen-Charpentier

López

Licea

et al. 2015

Mathematics and Computers in Simulation

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ElsevierChen Charpentier, BM.; Cortés, J.; Licea Salazar, JA.; Romero, J.; Roselló Ferragud, MD.; Santonja, F.; Villanueva Micó, RJ. (2015). Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach. AbstractDue to errors in measurements and inherent variability in the quantities of interest, models based on random differential equations give more realistic results than their deterministic counterpart. The generalized polynomial chaos (gPC) is a powerful technique used to approximate the solution of these equations when the random inputs follow standard probability distributions. But in many cases these random inputs do not have a standard probability distribution. In this paper, we present a step-by-step constructive methodology to implement directly a useful version of adaptive gPC for arbitrary distributions, extending the applicability of the gPC. The paper mainly focuses on the computational aspects, on the implementation of the method and on the creation of a useful software tool. This tool allows the user to easily change the types of distributions and the order of the expansions, and to study their the usefulness of the method are included.

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“…This property is fulfilled by probability distributions of uniform, Gaussian or beta type, for example. General distributions have to satisfy certain conditions, see [24], whereas a treatment of lognormal distributions can be found in [25]. Consequently, a function f ∈ L 2 (Π, ρ) exhibits a representation in the so-called polynomial chaos (PC), see [5],…”

Section: Polynomial Chaos Expansionmentioning

confidence: 99%

“…The quality of the MOR determines the magnitude of the term E 3 (s) in (24). Since the L 2 -norm of the probability space is employed, the MOR is required to be sufficiently accurate in subdomains of the parameters with relatively large probabilities.…”

Section: Error Analysismentioning

confidence: 99%

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

Pulch

Maten

2015

Int. J. UncertaintyQuantification

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Linear dynamical systems are considered in the form of ordinary differential equations or differential algebraic equations. We change their physical parameters into random variables to represent uncertainties. A stochastic Galerkin method yields a larger linear dynamical system satisfied by an approximation of the random processes. If the original systems own a high dimensionality, then a model order reduction is required to

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On the convergence of generalized polynomial chaos expansions

Cited by 362 publications

References 45 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

Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

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