Stability preservation in stochastic Galerkin projections of dynamical systems

TL;DR

This paper introduces a basis transformation technique that ensures stability preservation in stochastic Galerkin projections of dynamical systems, addressing instability issues in linear and nonlinear cases.

Contribution

It derives a basis transformation from Lyapunov equations that guarantees stability in stochastic Galerkin methods for dynamical systems.

Findings

Transformation guarantees stability in linear stochastic Galerkin systems

Preserves asymptotic stability of stationary solutions in nonlinear cases

Numerical examples confirm effectiveness of the approach

Abstract

In uncertainty quantification, critical parameters of mathematical models are substituted by random variables. We consider dynamical systems composed of ordinary differential equations. The unknown solution is expanded into an orthogonal basis of the random space, e.g., the polynomial chaos expansions. A Galerkin method yields a numerical solution of the stochastic model. In the linear case, the Galerkin-projected system may be unstable, even though all realizations of the original system are asymptotically stable. We derive a basis transformation for the state variables in the original system, which guarantees a stable Galerkin-projected system. The transformation matrix is obtained from a symmetric decomposition of a solution of a Lyapunov equation. In the nonlinear case, we examine stationary solutions of the original system. Again the basis transformation preserves the asymptotic stability of the stationary solutions in the stochastic Galerkin projection. We present results of numerical computations for both a linear and a nonlinear test example.