We propose a non-intrusive reduced-order modeling method based on Proper Orthogonal Decomposition (POD) and adaptive Multi-Element generalized Polynomial Chaos (ME-gPC) for efficient uncertainty quantification in parameterized problems. In the offline stage, a single or two-level POD extracts the spatial and/or temporal modes from selected full-order snapshots. The parameter-dependent coefficients are then approximated using ME-gPC, in which adaptive partitioning of the parameter space enables accurate local polynomial approximations, especially in low-regularity regions. The proposed approach is validated on three representative problems: a one-dimensional Burgers’ equation with a random force term, a flexible filament in a uniform flow, and the Kraichnan-Orszag three-mode problem. The results indicate that ME-gPC outperforms gPC in handling low-regularity issues, and the proposed method approximates the full-order model with high efficiency and accuracy. This demonstrates its strong potential for uncertainty quantification of space-time-dependent problems, particularly those with localized features, strong nonlinearity, or complex parameter interactions.
 

Reduced order modelling,Proper orthogonal decomposition (POD),Multi-Element generalized Polynomial Chaos,Space-time-dependent parameterized problems