Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

The accuracy and the computational e!ciency of a Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) method applied to stochastic problems with multiple uncertain input variables has been investigated. Two stochastic model problems with multiple uniform random variables were studied to determine the e"ect of di"erent sampling methods (Random, Latin Hypercube, and Hammersley) for the selection of the collocation points. The e" ect of the number of collocation points on the accuracy of polynomial chaos expansions were also investigated. The results of the stochastic model problems show that all three sampling methods exhibit a similar performance in terms of the the accuracy and the computational e!ciency of the chaos expansions. It has been observed that using a number of collocation points that is twice more than the minimum number required gives a better approximation to the statistics at each polynomial degree. This improvement can be related to the increase of the accuracy of the polynomial coe!cients due to the use of more information in their calculation. The results of the stochastic model problems also indicate that for problems with multiple random variables, improving the accuracy of polynomial chaos coe!cients in NIPC approaches may reduce the computational expense by achieving the same accuracy level with a lower order polynomial expansion. To demonstrate the application of Point-Collocation NIPC to an aerospace problem with multiple uncertain input variables, a stochastic computational aerodynamics problem which includes the numerical simulation of steady, inviscid, transonic flow over a three-dimensional wing with an uncertain free-stream Mach number and angle of attack has been studied. For this study, a 5 th degree Point-Collocation NIPC expansion obtained with Hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 Latin Hypercube Monte Carlo simulations with a significantly lower computational cost.

/pdf/efficient-sampling-for-non-intrusive-polynomial-chaos-4r4f9fdoc0.pdf

Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables

This paper describes a point-collocation nonintrusive polynomial chaos technique used for uncertainty propagation in computational fluid dynamics simulations. The application of point-collocation nonintrusive polynomial chaos to stochastic computational fluid dynamics is demonstrated with two examples: 1) a stochastic expansion-wave problem with an uncertain deflection angle (geometric uncertainty) and 2) a stochastic transonic-wing case with uncertain freestream Mach number and angle of attack. For each problem, input uncertainties are propagated with both the nonintrusive polynomial chaos method and Monte Carlo techniques to obtain the statistics of various output quantities. Confidence intervals for Monte Carlo statistics are calculated using the bootstrap method. For the expansion-wave problem, a fourth-degree polynomial chaos expansion, which requires five deterministic computational fluid dynamics evaluations, has been sufficient to predict the statistics within the confidence interval of 10,000 crude Monte Carlo simulations. In the transonic-wing case, for various output quantities of interest, it has been shown that a fifth-degree point-collocation nonintrusive polynomial chaos expansion obtained with Hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 Latin hypercube Monte Carlo simulations with a significantly lower computational cost. Overall, the examples demonstrate that the point-collocation nonintrusive polynomial chaos has a promising potential as an effective and computationally efficient uncertainty propagation technique for stochastic computational fluid dynamics simulations.

Point-Collocation Nonintrusive Polynomial Chaos Method for Stochastic Computational Fluid Dynamics

Bayesian estimates of parameter variability in the k-ε turbulence model

https://hal.archives-ouvertes.fr/hal-01200771/document

Predictive RANS simulations via Bayesian Model-Scenario Averaging

Complex o w and uid-structure interaction simulations require ecien t uncertainty quantication methods. In addition, a good property of an uncertainty quantication method is non-intrusiveness, meaning that existing deterministic solvers can be used for uncertainty quantication. In this paper the Probabilistic Collocation method is introduced, which combines the non-intrusiveness of the Stochastic Collocation method with the exponential convergence for arbitrary probability distributions of the Galerkin Polynomial Chaos method. Due to the non-intrusiveness and exponential convergence, the Probabilistic Collocation method requires only a few collocation points for a high accuracy. For the one dimensional piston problem the eciency of the Probabilistic Collocation method is compared with the Galerkin Polynomial Chaos method, the Non-Intrusive Polynomial Chaos method and the Stochastic Collocation method. The strength of the Probabilistic Collocation method is demonstrated by solving steady o w around a NACA0012 airfoil with an uncertain free stream velocity using a commercial o w solver. Dieren t possibilities of representing the stochastic solution are demonstrated to show the potential use of uncertainty quantication.

G.J.A. Loeven

Papers

Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties