On the guaranteed accuracy of Polynomial Chaos Expansions

This paper is concerned with the efficient simulation of stochastic nonlinear dynamical systems. A technique based on Polynomial Chaos Expansion (PCE) theory is used, in order to estimate the time evolution of the stochastic properties of the variables of interest. In PCE, each considered random variable is approximated by a truncated series of orthogonal polynomials, whose coefficients are identified by using the data collected in a relatively low number of numerical simulations. Then, the first and second order moments of the variables of interest, as well as an estimate of their probability density functions, can be efficiently recovered from the polynomial expansions. A least-squares identification approach is used here to identify the expansion's coefficients, and, in the framework of Set Membership identification theory, the issue of evaluating the guaranteed accuracy of the obtained PCE is tackled. As an example, the approach is tested on a nonlinear electric circuit.