2017/08/02 by Roland Pulch, Pulch, Roland, Florian Augustin +1
Decision Sciences · Engineering · #37H99 #65L20 #65L60 #FOS: Mathematics #Fault Detection and Control Systems #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1708.00958
openalex publication_date 2017/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In uncertainty quantification, critical parameters of mathematical models are\nsubstituted by random variables. We consider dynamical systems composed of\nordinary differential equations. The unknown solution is expanded into an\northogonal basis of the random space, e.g., the polynomial chaos expansions. A\nGalerkin method yields a numerical solution of the stochastic model. In the\nlinear case, the Galerkin-projected system may be unstable, even though all\nrealizations of the original system are asymptotically stable. We derive a\nbasis transformation for the state variables in the original system, which\nguarantees a stable Galerkin-projected system. The transformation matrix is\nobtained from a symmetric decomposition of a solution of a Lyapunov equation.\nIn the nonlinear case, we examine stationary solutions of the original system.\nAgain the basis transformation preserves the asymptotic stability of the\nstationary solutions in the stochastic Galerkin projection. We present results\nof numerical computations for both a linear and a nonlinear test example.\n