Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data

Ernst, Oliver G.; Powell, Catherine; Silvester, David J.; Ullmann, Elisabeth

doi:10.1137/070705817

Cited by 54 publications

(65 citation statements)

References 26 publications

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“…In that scenario, u and q denote the pressure and velocity field, respectively, and T is the permeability coefficient. In [11] and [15], T −1 is approximated directly by an M -term Karhunen-Loève (KL) expansion [21] which is a linear function of M uncorrelated random variables ξ m . In flow models, however, T often follows a lognormal distribution (e.g.…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…The differences between the saddle point systems in (3.1) and those encountered in previous work stem from the choice of approximation in (2.16). Suppose, as was assumed in [11] and [15], that we had started from a linear KL expansion…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…For (3.5) to satisfy (2.10), σ must be small relative to t 0 = T −1 and the term G 0 ⊗A 0 dominates in (3.6). The approximation A ≈ G 0 ⊗ A 0 was exploited in [11] to obtain a preconditioner of type (1.2). For stochastically linear problems, such mean-based approximations are effective within the regime of statistical parameters where the problem is well-posed.…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…In [11] and [15] an SG mixed formulation of the steady-state diffusion problem is studied and block-diagonal preconditioners for the resulting saddle point matrices are proposed. In those works, however, the diffusion coefficient is a linear function of M bounded random parameters, yielding N = M and well-conditioned, sparse matrices G 0 , G 1 , .…”

mentioning

confidence: 99%

“…Properties of the resulting saddle point matrices are discussed in Section 3. In Section 4 we study a Schur complement preconditioner from [11] and in Section 5 we revisit a preconditioner from [15], which is equivalent to P A in (1.3) with γ = 1 and a certain W . We make essential improvements to both preconditioners for the new model problem by combining them with best Kronecker product approximation (see [33]).…”

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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%

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients

Traverso

Phillips

Yang

2013

Numerical Methods in Fluids

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SUMMARYThis paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single-phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen-Loève expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss-Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen-Loève expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables.

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Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

Traverso

Phillips

2020

Numerical Methods in Fluids

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Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen-Loève expansion (KLE). The stochastic variability of permeability is modelled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss-Seidel used as a preconditioner for CG is shown to be efficient and robust for SFEM.

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Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data

Cited by 54 publications

References 26 publications

Preconditioning Stochastic Galerkin Saddle Point Systems

Preconditioning Stochastic Galerkin Saddle Point Systems

Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients

Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

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