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.

翻译：本文讨论基于时间无关线性和非线性偏微分方程系统的随机模型参数不确定性量化问题，采用无采样的侵入式随机伽辽金方法，并利用结合两重网格施瓦兹方法与代数多重网格构建的多层级可扩展求解器。高分辨率空间网格与大量随机展开项导致系统规模增大，引发显著的内存消耗与计算成本。为此，针对随机线性和非线性偏微分方程，开发了基于区域分解的并行可扩展求解器。构建了配备由精细网格受限加性施瓦兹预条件子和粗网格代数多重网格预条件子组成的多层级预条件器的广义最小残差迭代求解器，以提升可扩展性。数值实验展示了所提出求解器在随机线性和非线性泊松问题中的可扩展性能。