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.

中文翻译：

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基于B样条Bézier元素的方法和正确的正交分解的非侵入式降阶建模，用于求解时间相关问题的不确定性：在溃坝流中的应用

提出了一种适当的基于正交分解的B样条B'ezier元素方法（POD-BSBEM），作为用于随机时间相关问题的不确定性传播分析的非侵入式降阶模型。该方法使用两步适当的正交分解（POD）技术从称为快照的高保真解决方案集合中提取简化的基础。然后，将第三POD级别应用于与缩减基数相关联的投影系数的数据，以将与时间相关的模式与随机参数化系数分开。这些在随机参数空间中使用在相应的B'ezier元素中定义的B样条基函数进行近似。使用基准稳态和与时间有关的问题评估了该方法的准确性和效率，并将其与基于降阶模型的人工神经网络（POD-ANN）和基于全阶模型的多项式混沌扩展进行了比较（全PCE）。然后，将POD-BSBEM应用于分析复杂水深的河流中假设水坝溃决所产生的洪水流中的不确定性传播。结果证实了POD-BSBEM能够准确预测目标输出量的统计矩的能力，与其他技术相比，离线和在线阶段的速度都大大提高。然后，将POD-BSBEM应用于分析复杂水深的河流中假设水坝溃决所产生的洪水流中的不确定性传播。结果证实了POD-BSBEM能够准确预测目标输出量的统计矩的能力，与其他技术相比，离线和在线阶段的速度都大大提高。然后，将POD-BSBEM应用于分析复杂水深的河流中假设水坝溃决所产生的洪水流中的不确定性传播。结果证实了POD-BSBEM能够准确预测目标输出量的统计矩的能力，与其他技术相比，离线和在线阶段的速度都大大提高。