  This article discusses the uncertainty quantification (UQ) for
time-independent linear and nonlinear partial differential equation (PDE)-based
systems with random model parameters carried out using sampling-free intrusive
stochastic Galerkin method leveraging multilevel scalable solvers constructed
combining two-grid Schwarz method and AMG. High-resolution spatial meshes along
with a large number of stochastic expansion terms increase the system size
leading to significant memory consumption and computational costs. Domain
decomposition (DD)-based parallel scalable solvers are developed to this end
for linear and nonlinear stochastic PDEs. A generalized minimum residual
(GMRES) iterative solver equipped with a multilevel preconditioner consisting
of restricted additive Schwarz (RAS) for the fine grid and algebraic multigrid
(AMG) for the coarse grid is constructed to improve scalability. Numerical
experiments illustrate the scalabilities of the proposed solver for stochastic
linear and nonlinear Poisson problems.
