An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties. (English) Zbl 1275.65004
Summary: F. Augustin et al. [Eur. J. Appl. Math. 19, No. 2, 149–190 (2008; Zbl 1148.65004)] considered the polynomial chaos expansion for the treatment of uncertainties in industrial applications. For many applications the method has been proven to be a computationally superior alternative to Monte Carlo evaluations. In the current overview we compare the accuracy of polynomial chaos type methods for the propagation of uncertainties in nonlinear problems and verify them on two examples relevant for the industry. For weakly nonlinear time-dependent models, the generalized Kalman filter equations define an efficient method, yielding good approximations if the quantities of interest are restricted to the first two moments of the solution. Secondly, stochastic collocation is discussed. The method is applied to delay differential equations and random ordinary differential equations. Finally, a generalized polynomial chaos method is discussed which is based on a subdivision of the random space. This approach is even suitable for highly nonlinear models.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
polynomial chaos expansion; Monte Carlo; propagation of uncertanties; nonlinear time-dependent models; Kalman filter equation; stochastic collocation; delay differential equations; random ordinary differential equationsCitations:
Zbl 1148.65004References:
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