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Zbl 1052.65114
Le Maître, O.P.; Knio, O.M.; Najm, H.N.; Ghanem, R.G.
Uncertainty propagation using Wiener-Haar expansions. (English)
[J] J. Comput. Phys. 197, No. 1, 28-57 (2004). ISSN 0021-9991

Summary: An uncertainty quantification scheme is constructed based on generalized polynomial chaos 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. \par 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 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.

MSC 2000:: *65P20 Numerical chaos
65T60 Wavelets
37C75 Stability theory
37D45 Strange attractors, chaotic dynamics
65P40 Nonlinear stabilities
37M25 Computational methods for ergodic theory

Keywords: Wavelets; Polynomial Chaos; Stochastic process; Uncertainty quantification; Galerkin method; numerical examples; dynamical system; Rayleigh-Bénard instability; Wiener-Legendre decomposition; convergence

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