GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Probability (2nd ed.)
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Advances in Engineering Software - Special issue on large-scale analysis, design and intelligent synthesis environments
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Karhunen-Loève approximation of random fields by generalized fast multipole methods
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Computational aspects of the stochastic finite element method
Computing and Visualization in Science
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity
SIAM Journal on Numerical Analysis
Probabilistic models for stochastic elliptic partial differential equations
Journal of Computational Physics
A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
Sparse Tensor Discretization of Elliptic sPDEs
SIAM Journal on Scientific Computing
Preconditioning Stochastic Galerkin Saddle Point Systems
SIAM Journal on Matrix Analysis and Applications
IEEE Transactions on Information Theory
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We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a stochastic convection-diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated Gaussian random variables. The associated Galerkin matrix is nonsymmetric but sparse and allows for fast matrix-vector multiplications with optimal complexity. We construct and analyze two block-diagonal preconditioners for this Galerkin matrix for use with Krylov subspace methods such as the generalized minimal residual method. We test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection-diffusion formulations.