Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
type: Item
instance of: scholarly article
title: Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
zbMATH Open document ID: 1281.60054
author: Andrew J. Majda
author: Michal Branicki
published in: Communications in Mathematical Sciences
publication date: 2013-08-22T00:00:00Z
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 76F55
zbMATH DE Number: 6200795
zbMATH Keywords: polynomial chaos
zbMATH Keywords: uncertainty quantification
zbMATH Keywords: intermittency
zbMATH Keywords: white noise
zbMATH Keywords: parametric uncertainty
MaRDI profile type: Publication
full work available at URL: cms.2013.v11.n1.a3
OpenAlex ID: W2063334232
DOI: 10.4310/CMS.2013.V11.N1.A3
review text: 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.
description: scientific article; zbMATH DE number 6200795