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.

翻译：在数学模型中引入概率项对于捕捉和量化真实世界系统中的不确定性至关重要。实际上，随机性可能对问题解的行为产生显著影响，因此需要更深入的分析以获得更现实且更具信息量的结果。另一方面，由于需要生成大量系统实现以获得有意义的统计数据，随机模型的研究可能需要大量计算资源。这使得开发复杂度降低技术（如替代模型）对于实现高效且可扩展的模拟至关重要。在本研究中，我们利用多项式混沌展开来研究流体动力学中分岔现象（即科恩达效应）的替代表示的准确性——在随机设定下，该现象为解的非唯一性提供了不同视角。随后，我们描述了如何将其纳入有限元框架，从而得出增强型谱随机有限元方法的公式。此外，我们探究了确定性分岔图与多项式混沌多项式之间的联系，揭示了它们在重构整个解流形方面的能力。