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Uncertainty propagation using Wiener-Haar expansions. (English) Zbl 1052.65114
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.
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:
65P20 Numerical chaos
65T60 Numerical methods for wavelets
37C75 Stability theory for smooth dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65P40 Numerical nonlinear stabilities in dynamical systems
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
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