TY - JOUR
T1 - Stochastic Galerkin Methods for Linear Stability Analysis of Systems with Parametric Uncertainty
AU - Sousedík, Bedřich
AU - Lee, Kookjin
N1 - Funding Information:
∗Received by the editors April 27, 2021; accepted for publication (in revised form) March 2, 2022; published electronically September 27, 2022. https://doi.org/10.1137/21M1415595 Funding: This work was supported by the U. S. National Science Foundation under grant DMS-1913201. †Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 USA (sousedik@umbc.edu). ‡The School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ 85281 USA (klee263@asu.edu).
Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - Navier-Stokes equation
KW - eigenvalue analysis
KW - linear stability
KW - preconditioning
KW - spectral stochastic finite element method
KW - stochastic Galerkin method
KW - uncertainty quantification
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U2 - 10.1137/21M1450392
DO - 10.1137/21M1450392
M3 - Article
AN - SCOPUS:85140140887
VL - 10
SP - 1101
EP - 1129
JO - SIAM-ASA Journal on Uncertainty Quantification
JF - SIAM-ASA Journal on Uncertainty Quantification
SN - 2166-2525
IS - 3
ER -