Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
This paper studies truncated polynomial chaos expansions as possible methods to determine evolution of uncertainty in nonlinear dynamical systems with intermittency or fat-tail probabilty densities. In particular, truncated polynomial chaos expansions (PCE) are studied together with truncated eigenfunction expansions of the joint probability density satisfying a corresponding Fokker-Planck equation, these expansions are referred to as Gram-Charlier expansions (GrChE).NEWLINENEWLINEThe authors present a generally nonlinear non-Gaussian test model described by two coupled Itō stochastic differential equationsNEWLINENEWLINENEWLINE\[NEWLINE \begin{aligned} du(t) &= (-(\hat\gamma+a_{u}\gamma(t))u(t)+f(t) )\,dt + \sigma_{u}\,dW_{u}(t),\\ d\gamma(t) &= -(d_\gamma+a_\gamma u(t))\gamma(t) \,dt + \sigma_\gamma\,dW_\gamma(t), \end{aligned} NEWLINE\]NEWLINENEWLINENEWLINEwhere \(\hat\gamma, d_\gamma, \sigma_{u}, \sigma_\gamma > 0\) are positive constants and \(W_{u}\) and \(W_\gamma\) are two independent real-valued Wiener processes. Different choices of model parameters are presented in four particular cases of different dynamical behaviour. The above test model is used to show that both PCE and GrChE have serious fundamental limitations arising not only from their finite truncation.
scientific article; zbMATH DE number 6200795