Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties of real-world systems. However, stochastic models typically require large computational resources to produce meaningful statistics. For such reason, the development of reduction techniques becomes essential for enabling efficient and scalable simulations of complex scenarios while quantifying the underlying uncertainties. In this work, we study the accuracy of Polynomial Chaos (PC) surrogate expansion of the probability space on a bifurcating phenomena in fluid dynamics, namely the Coand\u{a} effect. In particular, we propose a novel non-deterministic approach to generic bifurcation problems, where the stochastic setting gives a different perspective on the non-uniqueness of the solution, also avoiding expensive simulations for many instances of the parameter. Thus, starting from the formulation of the Spectral Stochastic Finite Element Method (SSFEM), we extend the methodology to deal with solutions of a bifurcating problem, by working with a perturbed version of the deterministic model. We discuss the link between the deterministic and the stochastic bifurcation diagram, highlighting the surprising capability of PC polynomials coefficients of giving insights on the deterministic solution manifold.

翻译：在数学模型中引入概率项对于捕捉和量化现实世界系统的不确定性至关重要。然而，随机模型通常需要大量计算资源才能产生有意义的统计结果。因此，发展降阶技术对于实现复杂场景的高效可扩展模拟，同时量化潜在不确定性变得至关重要。本研究探讨了概率空间的多项式混沌（PC）代理展开在流体动力学分岔现象（即柯恩达效应）上的准确性。特别地，我们针对一般分岔问题提出了一种新颖的非确定性方法，其中随机设定为解的非唯一性提供了不同视角，同时避免了对参数多个实例进行昂贵模拟。基于谱随机有限元法（SSFEM）的表述，我们通过处理确定性模型的扰动版本，将该方法扩展到分岔问题的求解。我们讨论了确定性分岔图与随机分岔图之间的联系，并揭示了PC多项式系数在揭示确定性解流形特性方面令人惊讶的能力。