Stability preservation in stochastic Galerkin projections of dynamical systems
This work provides a solution to a known instability issue in uncertainty quantification for dynamical systems, which is important for practitioners using polynomial chaos expansions.
The authors address instability in stochastic Galerkin projections of dynamical systems and derive a basis transformation that guarantees stability preservation for both linear and nonlinear cases, demonstrated with numerical examples.
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.