Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty

TL;DR

This paper introduces a stochastic Galerkin approach for linear stability analysis of PDE systems with uncertain parameters, focusing on efficient eigenvalue computation for nonsymmetric operators, and compares its performance to Monte Carlo methods.

Contribution

It develops an efficient inexact Newton-GMRES method for stochastic eigenvalue problems with nonsymmetric operators, applied to Navier-Stokes equations with stochastic viscosity.

Findings

The method accurately characterizes the rightmost eigenvalue under uncertainty.

It outperforms Monte Carlo and stochastic collocation in efficiency.

Numerical experiments demonstrate the method's effectiveness.

Abstract

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 Navier-Stokes equation with stochastic viscosity, its accuracy is compared to that of Monte Carlo and stochastic collocation, and the efficiency is illustrated by numerical experiments.