A stochastic perturbation approach to nonlinear bifurcating problems

Abstract

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.

Figures (20)

• Figure 1: Statistical analysis of bifurcations via the stochastic-perturbation approach.
• Figure 2: Synthetic workflow diagram showing the process of computing the corresponding PDF of $u(\bm{x},\omega)$ by solving Equation \ref{['eq:general_prob']} using SSFEM.
• Figure 3: Some admissible PC solutions with $N_{PC}=5$ for large and small perturbations of $\mu$, respectively top and bottom, considering a Gaussian and a Uniform distribution, respectively left and right. On top, the sampling is agnostic of the bifurcating region, on bottom it is located in the previously identified regime.
• Figure 4: Statistics on the PDFs computed through KDE for $100$ random initializations of the solver with $N_{PC}=5$, for $\mu\sim\mathcal{N}(1,0.06)$ and $\mu\sim\mathcal{U}(0.8,1.2)$, left and right respectively.
• Figure 5: Comparison between deterministic bifurcation diagram and the solution PDFs computed for $500$ different uniform perturbations of $\mu$, with constant low variance and different mean $\mu\sim\mathcal{U}(\overline{\mu}-0.01,\overline{\mu}+0.01)$ with $\overline{\mu} \in [-0.5,1.5]$.