Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems. Indeed, randomness can have a significant impact on the behavior of the problem's solution, and a deeper analysis is needed to obtain more realistic and informative results. On the other hand, the investigation of stochastic models may require great computational resources due to the importance of generating numerous realizations of the system to have meaningful statistics. This makes the development of complexity reduction techniques, such as surrogate models, essential for enabling efficient and scalable simulations. In this work, we exploit polynomial chaos (PC) expansion to study the accuracy of surrogate representations for a bifurcating phenomena in fluid dynamics, namely the Coanda effect, where the stochastic setting gives a different perspective on the non-uniqueness of the solution. Then, its inclusion in the finite element setting is described, arriving to the formulation of the enhanced Spectral Stochastic Finite Element Method (SSFEM). Moreover, we investigate the connections between the deterministic bifurcation diagram and the PC polynomials, underlying their capability in reconstructing the whole solution manifold.

中文翻译：

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非线性分岔问题的随机扰动方法

将概率项纳入数学模型对于捕获和量化现实系统中的不确定性至关重要。事实上，随机性会对问题解决方案的行为产生重大影响，需要进行更深入的分析才能获得更现实和信息丰富的结果。另一方面，由于生成系统的大量实现以获得有意义的统计数据的重要性，随机模型的研究可能需要大量的计算资源。这使得开发降低复杂性的技术（例如代理模型）对于实现高效且可扩展的模拟至关重要。在这项工作中，我们利用多项式混沌（PC）展开来研究流体动力学中分叉现象（即康达效应）的替代表示的准确性，其中随机设置对解决方案的非唯一性给出了不同的视角。然后，描述了其在有限元设置中的包含情况，得出增强谱随机有限元法（SSFEM）的公式。此外，我们研究了确定性分岔图和 PC 多项式之间的联系，这是它们重建整个解流形的能力的基础。