Speaker
Description
Incorporating probabilistic terms into mathematical models is essential for Uncertainty Quantification in complex systems. However, standard stochastic methods, such as Monte Carlo simulations, are often computationally intensive, particularly when investigating ill-conditioned problems such as bifurcating phenomena in parameter-dependent PDEs, which require running a large number of simulations across different parameter values. In contrast, this work presents a Spectral Stochastic Finite Element Method (SSFEM) framework, based on generalized Polynomial Chaos Expansion (PCE), that captures system properties while requiring only a few runs of the nonlinear solver for the resulting Galerkin system.
The proposed approach analyzes a perturbed version of the original problem, employing PC coefficients to reconstruct bifurcation diagrams without the need for extensive parameter sampling. The methodology is grounded in the theoretical analysis of one-dimensional normal forms in dynamical systems, where we characterize two distinct solution families: oscillating solutions, which emerge when multiple stable states coexist, and branch-approximating solutions, which track specific bifurcation branches. Together, these provide both analytical convergence results and statistical properties that enable the identification of bifurcation branch locations.
We extend this surrogate methodology to high-dimensional PDEs, specifically applied to problems in continuum mechanics. The results demonstrate that PC coefficients yield significant statistical information regarding the solution manifold, offering a scalable path for the high-order numerical analysis of complex phenomena.