Uncertainty propagation using Wiener–Haar expansions

An uncertainty quantification scheme is constructed based on generalized Polynomial Chaos (PC) representations. Two such representations are considered, based on the orthogonal projection of uncertain data and solution variables using either a Haar or a Legendre basis. Governing equations for the unknown coefficients in the resulting representations are derived using a Galerkin procedure and then integrated in order to determine the behavior of the stochastic process. The schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh–Bénard instability. For situations involving random parameters close to a critical point, the computational implementations show that the Wiener–Haar (WHa) representation provides more robust predictions that those based on a Wiener–Legendre (WLe) decomposition. However, when the solution depends smoothly on the random data, the WLe scheme exhibits superior convergence. Suggestions regarding future extensions are finally drawn based on these experiences.