[Paper Review] A stochastic perturbation approach to nonlinear bifurcating problems

Isabella Carla Gonnella, Moaad Khamlich+2 more|arXiv (Cornell University)|Feb 26, 2024

Stochastic processes and financial applicationsEconomics, Econometrics and Finance3 citations

TL;DR

This paper proposes a stochastic perturbation approach using Polynomial Chaos (PC) expansion within the Spectral Stochastic Finite Element Method (SSFEM) to characterize solution branches in nonlinear bifurcating problems, such as the Coanda effect in fluid dynamics. It demonstrates that local extrema in PC surrogate functions directly correspond to the number and values of coexisting solution branches, enabling reconstruction of the full solution manifold without prior knowledge of bifurcation topology.

ABSTRACT

Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems, especially when the solution is not unique or exhibits sudden qualitative changes as parameters vary. However, stochastic models typically require large computational resources to produce meaningful statistics. In this work, we leverage the Polynomial Chaos (PC) expansion to propose a systematic approach for bifurcation detection in parametric systems of equations. We show that the method, exploiting a perturbed version of the deterministic model, avoids repeated costly simulations across multiple parameter values and requires no prior information for initializing numerical solvers, while still providing accurate characterization of the bifurcation branches. We argue that the PC solutions of the perturbed model not only provide access to statistical information about the deterministic branches, but also approximate these branches in a meaningful sense. Finally, we validate our claims by means of numerical tests on the pitchfork bifurcation, examining both its normal form and a classical realization in fluid-dynamics PDEs, namely the Coanda effect.

Motivation & Objective

• To address the computational intractability of reconstructing full bifurcation diagrams using traditional deterministic continuation methods across large parametric ranges.
• To investigate whether Polynomial Chaos (PC) surrogate models can accurately encode the topology and values of coexisting solution branches in nonlinear bifurcating systems.
• To develop an intrusive SSFEM framework that enables efficient, high-fidelity stochastic analysis of bifurcating phenomena without requiring problem-specific initialization.
• To establish a quantitative link between the number and location of local extrema in PC expansions and the number and values of solution branches in the deterministic bifurcation diagram.
• To enable probabilistic reconstruction of the bifurcation diagram through statistical analysis of SSFEM solutions, particularly in regimes of non-uniqueness.

Proposed method

• Model the bifurcation parameter (e.g., viscosity) as a stochastic process using Karhunen-Loève (K-L) expansion to represent its uncertainty.
• Represent the solution of the stochastic PDE using a Polynomial Chaos (PC) expansion in terms of orthogonal polynomials, with K-L eigenfunctions as random variables.
• Formulate an intrusive SSFEM approach by projecting the weak form of the PDE onto the PC basis, resulting in a coupled system of equations for the PC coefficients.
• Solve the resulting high-dimensional system using a nonlinear solver to compute the PC coefficients, enabling statistical analysis via orthogonal properties of PC polynomials.
• Reconstruct the solution manifold by sampling the PC expansion and analyzing the distribution of solution values, particularly identifying clusters corresponding to solution branches.
• Compare the probabilistic solution structure (evidenced by local extrema and density clusters) with the deterministic bifurcation diagram to validate the method’s accuracy.

Experimental results

Research questions

• RQ1 Can Polynomial Chaos expansions of the solution in a stochastic setting accurately reconstruct the number and values of coexisting solution branches in a bifurcating system?
• RQ2 How do the local extrema in the PC surrogate functions relate to the branches of the deterministic bifurcation diagram?
• RQ3 To what extent can the SSFEM framework reconstruct the full solution manifold without prior knowledge of the bifurcation topology?
• RQ4 How does the stochastic perturbation approach compare to traditional deterministic continuation methods in terms of computational efficiency and robustness across parametric ranges?
• RQ5 Can the probabilistic solution structure (e.g., density clusters) reliably indicate the presence and stability of solution branches in non-unique regimes?

Key findings

• The number of local extrema in the PC surrogate function within the sampling region matches the number of coexisting solution branches for the corresponding deterministic parameter value.
• The values of the local extrema in the PC expansion correspond closely to the actual values of the solution branches in the deterministic bifurcation diagram.
• For viscosity values such as μ = 0.9, the probabilistic solution via SSFEM successfully recovers both branches of the deterministic bifurcation diagram.
• At μ = 1.1, where only one solution exists, the solution density concentrates on a single value, confirming the absence of multiple branches.
• Even in strong bifurcation regimes (e.g., μ = 0.85 and μ = 0.8), the evolution of the branches is captured, though convergence issues slightly distort the symmetry of the lower branch.
• The intrusive SSFEM approach enables full reconstruction of the solution manifold without requiring problem-specific initialization or prior knowledge of bifurcation behavior.