  Stochastic Galerkin finite element method (SGFEM) provides an efficient
alternative to traditional sampling methods for the numerical solution of
linear elliptic partial differential equations with parametric or random
inputs. However, computing stochastic Galerkin approximations for a given
problem requires the solution of large coupled systems of linear equations.
Therefore, an effective and bespoke iterative solver is a key ingredient of any
SGFEM implementation. In this paper, we analyze a class of truncation
preconditioners for SGFEM. Extending the idea of the mean-based preconditioner,
these preconditioners capture additional significant components of the
stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a
model problem and assuming affine-parametric representation of the diffusion
coefficient, we perform spectral analysis of the preconditioned matrices and
establish optimality of truncation preconditioners with respect to SGFEM
discretization parameters. Furthermore, we report the results of numerical
experiments for model diffusion problems with affine and non-affine parametric
representations of the coefficient. In particular, we look at the efficiency of
the solver (in terms of iteration counts for solving the underlying linear
systems) and compare truncation preconditioners with other existing
preconditioners for stochastic Galerkin matrices, such as the mean-based and
the Kronecker product ones.
