Stochastic Galerkin Methods for Linear Stability Analysis of Systems with Parametric Uncertainty

Bedřich Sousedík, Kookjin Lee

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the form of generalized polynomial chaos expansion. The stability analysis leads to the solution of a stochastic eigenvalue problem, and we wish to characterize the rightmost eigenvalue. We focus, in particular, on problems with nonsymmetric matrix operators, for which the eigenvalue of interest may be a complex conjugate pair, and we develop methods for their efficient solution. These methods are based on inexact, line-search Newton iteration, which entails use of preconditioned GMRES. The method is applied to linear stability analysis of the Navier{Stokes equations with stochastic viscosity, its accuracy is compared to that of Monte Carlo and stochastic collocation, and the efficiency is illustrated by numerical experiments.

Original languageEnglish (US)
Pages (from-to)1101-1129
Number of pages29
JournalSIAM-ASA Journal on Uncertainty Quantification
Issue number3
StatePublished - 2022


  • Navier-Stokes equation
  • eigenvalue analysis
  • linear stability
  • preconditioning
  • spectral stochastic finite element method
  • stochastic Galerkin method
  • uncertainty quantification

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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