Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Stochastic analysis
Preconditioning in H(div) and applications
Mathematics of Computation
Iterative methods for solving linear systems
Iterative methods for solving linear systems
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Optimal Preconditioning for Raviart--Thomas Mixed Formulation of Second-Order Elliptic Problems
SIAM Journal on Matrix Analysis and Applications
Karhunen-Loève approximation of random fields by generalized fast multipole methods
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Preconditioning and convergence in the right norm
International Journal of Computer Mathematics - Fast Iterative and Preconditioning Methods for Linear and Non-Linear Systems
SIAM Journal on Scientific Computing
Iterative Solvers for the Stochastic Finite Element Method
SIAM Journal on Scientific Computing
A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
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Mixed finite element discretizations of deterministic second-order elliptic 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 nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (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.