Optimal and superoptimal circulant preconditioners
SIAM Journal on Matrix Analysis and Applications
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
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.)
A review of algebraic multigrid
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Multigrid
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation
Journal of Scientific Computing
Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure
SIAM Journal on Scientific Computing
Journal of Computational Physics
Numerical Methods for Differential Equations in Random Domains
SIAM Journal on Scientific Computing
Stochastic analysis of transport in tubes with rough walls
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
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This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs. Specialized solvers are required to solve the very high-dimensional systems that result after a finite element discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; and multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based preconditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties.