  In this work, solvers and preconditioners based on Gauss-Seidel and Jacobi
algorithms are explored for stochastic Galerkin discretization of partial
differential equations (PDEs) with random input data. Gauss-Seidel and Jacobi
algorithms are formulated such that the existing software is leveraged in the
computational effort. These algorithms are also used as preconditioners to
Krylov iterative methods. The solvers and preconditiners are compared with
Krylov based iterative methods with the traditional mean-based preconditioner
[13] and Kronecker-product preconditioner [17] by solving a steady state state
advection-diffusion equation, which upon discretization, results in a
non-symmetric positive definite matrix on left-hand-side. Numerical results
show that an approximate version of Gauss-Seidel algorithm is a good
preconditioner for GMRES to solve non-symmetric Galerkin system of equations.
