  A proper orthogonal decomposition-based B-splines B\'ezier elements method
(POD-BSBEM) is proposed as a non-intrusive reduced-order model for uncertainty
propagation analysis for stochastic time-dependent problems. The method uses a
two-step proper orthogonal decomposition (POD) technique to extract the reduced
basis from a collection of high-fidelity solutions called snapshots. A third
POD level is then applied on the data of the projection coefficients associated
with the reduced basis to separate the time-dependent modes from the stochastic
parametrized coefficients. These are approximated in the stochastic parameter
space using B-splines basis functions defined in the corresponding B\'ezier
element. The accuracy and the efficiency of the proposed method are assessed
using benchmark steady-state and time-dependent problems and compared to the
reduced order model-based artificial neural network (POD-ANN) and to the
full-order model-based polynomial chaos expansion (Full-PCE). The POD-BSBEM is
then applied to analyze the uncertainty propagation through a flood wave flow
stemming from a hypothetical dam-break in a river with a complex bathymetry.
The results confirm the ability of the POD-BSBEM to accurately predict the
statistical moments of the output quantities of interest with a substantial
speed-up for both offline and online stages compared to other techniques.
