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Stochastic Galerkin finite element method (SGFEM) provides an efficientalternative to traditional sampling methods for the numerical solution oflinear elliptic partial differential equations with parametric or randominputs. However, computing stochastic Galerkin approximations for a givenproblem requires the solution of large coupled systems of linear equations.Therefore, an effective and bespoke iterative solver is a key ingredient of anySGFEM implementation. In this paper, we analyze a class of truncationpreconditioners for SGFEM. Extending the idea of the mean-based preconditioner,these preconditioners capture additional significant components of thestochastic Galerkin matrix. Focusing on the parametric diffusion equation as amodel problem and assuming affine-parametric representation of the diffusioncoefficient, we perform spectral analysis of the preconditioned matrices andestablish optimality of truncation preconditioners with respect to SGFEMdiscretization parameters. Furthermore, we report the results of numericalexperiments for model diffusion problems with affine and non-affine parametricrepresentations of the coefficient. In particular, we look at the efficiency ofthe solver (in terms of iteration counts for solving the underlying linearsystems) and compare truncation preconditioners with other existingpreconditioners for stochastic Galerkin matrices, such as the mean-based andthe Kronecker product ones.

Truncation preconditioners for stochastic Galerkin finite element discretizations