Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems. However, stochastic models typically require large computational resources to produce meaningful statistics. For this 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 phenomenon in fluid dynamics, namely the Coanda effect. In particular, we propose a novel non-deterministic approach to generic bifurcation detection problems, where the stochastic setting provides 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 deterministic and stochastic bifurcation diagrams, highlighting the surprising capability of PC polynomial coefficients to give insights into 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中的不确定性至关重要。然而，随机模型通常需要大量的计算资源才能产生有意义的统计结果。因此，开发降阶技术对于实现对复杂场景的高效、可扩展模拟，同时量化潜在的不确定性，变得至关重要。在本研究中，我们探讨了概率空间的混沌多项式（PC）代理展开在流体动力学分岔现象（即科恩达效应）上的准确性。特别地，我们针对通用分岔检测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的视角，同时也避免了对参数众多实例进行昂贵的模拟。因此，从谱随机有限元法（SSFEM）的公式出发，我们通过处理确定性模型的扰动版本，将方法论扩展到处理分岔问题的解。我们讨论了确定性与随机分岔图之间的联系，强调了PC多项式系数在揭示确定性解流形方面出人意料的能力。