Iterative Solvers for the Stochastic Finite Element Method

Rosseel, Eveline; Vandewalle, Stefan

doi:10.1137/080727026

Cited by 55 publications

(96 citation statements)

References 34 publications

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“…G 0 ⊗ A 0 is not a good approximation to A due, in part, to a dramatic increase in the ill-conditioning of the G n matrices. For SG finite element discretizations of (2.2), preconditioners have been suggested in [30] and [32]. SG methods can only be competitive for challenging PDEs, however, if robust preconditioners are found for the coupled linear systems of equations they yield.…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…, G N represent multiplication operators on a probability space associated with the random PDE coefficients. Their structural and spectral properties (see [12], [27], [30]) are governed by our choice of discretization on the probability space. We assume that the (1,1) block in (1.4) is positive definite and so linear systems with this C can be solved via preconditioned minres with the block-diagonal preconditioners described above.…”

mentioning

confidence: 99%

“…Stochastic Galerkin approximation, specifically, has been studied in [4] and [9] and solvers for the resulting symmetric positive definite linear systems have been studied in [27], [32] and [30].…”

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confidence: 99%

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

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Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

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confidence: 99%

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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“…the system (19) can be rewritten taking advantage of the Kronecker product [46]. This representation of the reluctivity as a sum of separable functions can be obtained either during the process of probabilistic modelling of the input data or by applying a model reduction technique (Karuhnen-Loeve expansion for example).…”

Section: Galerkin Methods : Stochastic Finite Element 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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“…The matrices D i , G 2 ∈ R Q×Q characterize the stochastics of the problem and are respectively defined by D i = Ψ i ΨΨ T and G 2 = ξ 2 ΨΨ T . In [26,10] properties of these matrices are given. Each individual matrix is a sparse matrix, however the sum, …”

Section: Newton Linearizationmentioning

confidence: 99%

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Rosseel

Gersem

Vandewalle

2010

CiCP

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The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost. AbstractThe stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.

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Iterative Solvers for the Stochastic Finite Element Method

Cited by 55 publications

References 34 publications

Preconditioning Stochastic Galerkin Saddle Point Systems

Preconditioning Stochastic Galerkin Saddle Point Systems

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

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