Block-diagonal preconditioning for spectral stochastic finite-element systems

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ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, background): “See for instance the works [3, 22, 23] for further details on space discretization and on the numerical solution of such system of equations.”
ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA (2011, background): “Again, problems to the right of the double line are ill-posed, and the Galerkin linear system as well as a subset of the individual collocation systems are guaranteed to become indefinite as the degree of polynomial approximation p (for stochastic Galerkin) or sparse grid level (for collocation) inc”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, background): “It has been shown in [19, 7] that the condition number of the preconditioned matrix is independent of the discretization parameters both in the physical and stochastic spaces, provided that uniform coercivity is ensured.”
A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations (2010, methodology): “In other words , P , Keese [ 17 ] , Powell and Elman [ 25 ] for random diffusion coefficients given in terms of a truncated KL expansion .”