Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

Abstract

The focus of this paper is on the use of polynomial chaos (PC) for developing surrogate models for differential algebraic equations (DAEs) with time-invariant uncertainties. Intrusive and nonintrusive approaches to synthesize PC surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations (ODEs), if the algebraic constraints are a function of the stochastic variable, some initial conditions of the DAEs are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a PC-based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the PC models are validated using Monte Carlo (MC) simulations in addition to estimating the evolving probably density functions (PDFs) of the states of the pendulum.