Block-diagonal preconditioning for spectral stochastic finite-element systems

Powell, Catherine; Elman, Howard C.

doi:10.1093/imanum/drn014

Cited by 147 publications

(268 citation statements)

References 24 publications

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“…The eigenvalues of the matrices G n in (3.7a) are known explicitly (see [27], [12]). For Gaussian random variables, the condition number of each G n grows, at worst, like O( √ d).…”

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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“…The eigenvalues of the matrices G n in (3.7a) are known explicitly (see [27], [12]). For Gaussian random variables, the condition number of each G n grows, at worst, like O( √ d).…”

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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“…See e.g. [3,17,18] for further details on space discretization and on the numerical solution of the system of equations.…”

Section: Stochastic Galerkin Methodsmentioning

confidence: 99%

Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients

et al. 2011

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Abstract. In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product.We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids.Key words: Uncertainty Quantification, PDEs with random data, elliptic equations, multivariate polynomial approximation, Best M -Terms approximation, Stochastic Galerkin methods, Smolyak approximation, Sparse grids, Stochastic Collocation methods.AMS Subject Classification: 41A10, 65C20, 65N12, 65N35 IntroductionMany works have been recently devoted to the analysis and the improvement of the Stochastic Galerkin and Collocation techniques for Uncertainty Quantification for PDEs with random input data. These methods are promising since they can exploit the possible regularity of the solution with respect to the stochastic parameters to achieve much faster convergence than sampling methods like Monte Carlo.Stochastic Galerkin and Collocation methods can be classified as parametric techniques, since both expand u, the solution of the PDE of interest, as a summation over suitable deterministic basis functions in probability space, typically polynomials or piecewise polynomials. Stochastic Galerkin is a projection technique over a set of orthogonal polynomials with respect to the probability measure at hand (see e.g. [1,12,15,20]), while Stochastic Collocation is a sum of Lagrangian interpolants over the probability space (see e.g. [2,10,22] 11The comparison between performances of these deterministic methods is a matter of study (see e.g.[3]). However, both suffer the so-called "Curse of Dimensionality": using naive projections/interpolations over tensor product polynomials spaces/tensor grids leads to computational costs that grow exponentially fast with the number of input random variables. In such a case, careful construction of approximation spaces/sparse grids is needed in order to retain accuracy while keeping computational work acceptably low.In a Stochastic Galerkin setting this requirement can be translated to the implementation of algorithms able to compute what is known as "Best M -Terms approximation". In other words, the method should be able to establish a-priori the set of the M most fruitful multivariate polynomials in the spectral approximation of the solution u, and to compute only those terms.Important contributions in the study of the Best M -Terms approximation have been given by Cohen, DeVore and Schwab: estimates on the decay of the coefficients of the spectral expansi...

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“…The second class of techniques relies on the model equations to derive a problem for the expansion coefficients through Galerkin-type procedures. It yields accurate solutions but usually requires the resolution of a large set of equations calling for ad hoc numerical strategies, such as Krylov type iterations [12,32,15] and preconditioning techniques [33,21], as well as an adaptation of the deterministic codes. The method presented in this paper focuses on the minimization of the computational cost in Galerkin methods for non-linear models.…”

Section: Introductionmentioning

confidence: 99%

Generalized spectral decomposition for stochastic nonlinear problems

Nouy

Maı̂tre

2009

Journal of Computational Physics

121

127

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We present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space.

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Block-diagonal preconditioning for spectral stochastic finite-element systems

Cited by 147 publications

References 24 publications

Preconditioning Stochastic Galerkin Saddle Point Systems

Preconditioning Stochastic Galerkin Saddle Point Systems

Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients

Generalized spectral decomposition for stochastic nonlinear problems

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