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Efficient collocational approach for parametric uncertainty analysis. (English) Zbl 1164.65302

Summary: A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by the polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounds is presented, along with numerical examples for problems with relatively complicated forms of governing equations.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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