Combined parametric–nonparametric uncertainty quantification using random matrix theory and polynomial chaos expansion

Propagation of combined parametric and nonparametric uncertainties in elliptic partial differential equations is considered. Two cases, namely, (a) both uncertainties are over the entire domain, and (b) different types of uncertainties are over non-overlapping subdomains are proposed. Parametric uncertainty is modelled by a random field and is discretised using the Karhunen–Loève (KL) expansion. The nonparametric uncertainty is modelled by Wishart random matrix. Both uncertainties are considered independent, and the two first moments of the response are calculated using polynomial chaos expansion and analytical random matrix theory results. Closed-form analytical expressions of the first two moments are derived for both cases.

► Parametric and nonparametric uncertainties in differential equations are considered.
► The uncertainties are modelled respectively with Random fields and Wishart matrices.
► Different uncertainties covering the same or different domains are discussed.
► New closed-form analytical expressions are derived to obtain response statistics.
► Direct Monte Carlo simulation and Numerical results show excellent agreement.