A Stochastic Collocation Algorithm for Uncertainty Analysis

Abstract This report describes a stochastic collocation method to adequately handle a physically intrinsic uncertaintyin the variables of a numerical simulation. For instance, while the standard Galerkin approach to PolynonfialChaos requires multi-dimensional summations over the stochastic basis functions, the stochastic collocationmethod enables to collapse those summations to a one-dimensional summation only. This report furnishes theessential algorithmic details of the new stochastic collocation method and provides as a numerical examplethe solution of the Riemann problem with the stochastic collocation method used for the discretization ofthe stochastic parameters. 1 Background With the continuous development and improvement of both CFD sinmlations reliability and accuracy, furtherdevelopments toward physically relevant results raise the need to account for more realistic operating condi-tions. Past simulations used deterministic parameters such as for the boundary conditions, initial conditions,geometry of the problem, physical properties, etc. Meanwhile, the actual conditions are not known preciselyand one must account for a certain level in uncertainty in the simulations. In a more general framework, itis indeed of critical importance to quanti_" uncertainty (e.g. [9, 1, 17]) and establish a confidence intervalfor the simulation-based predictions or design ([3 0. Unlike reliability analysis which deals with rare butcatastrophic events, uncertainty analysis is concerned with the probabilistic aspects of the simulations dueto the stochastic nature of physical parameters, initial and boundary conditions. Some sources of uncertaintyidentified in numerical simulations include:• inaccuracy / indeterminacy of initial conditions (experimental data, variability of operating environ-ment),• bias in physical models (turbulence model),• approximations in mathematical model (due to linearization or assumptions that neglect some physicaleffects such as temperature and/or pressure dependence of relevant physical parameters),• uncertainty in describing the physical reality (errors in geometry, roughness, boundary conditions),• discretization errors (numerical diffusion, round-off, algorithmic errors due to incomplete iterationprocess, etc.). See [1] or [41 for a more complete review.One crucial issue is to accurately propagate uncertainty from, say, the boundary conditions to the bulkfield and finally to the output of the simulation. Several techniques have been proposed in the past. Amongthese, the currently most commonly used if the so-called perturbation technique that basically involves ex-panding the variables of the problem in terms of Taylor series around their mean value (e.g. Ill D. Thougheffective, this technique is limited to Gaussian or weakly non-Gaussian processes due to the difficulty inincorporating terms of order higher than two. Still, this technique allows for a quick estimation of the loworder statistics.When one wants to get access to the full statistics, one can make use of the Monte Carlo technique. Itis considered as an "exact" method for accounting for uncertainty in the sense that it does not require anyapproximations nor assumptions. The main advantages are that its convergence rate does not depend on thenumber of independent random variables and that its application is straightforward. However, it becomesintractable for large problems as it requires to carry thousands of simulations which results in prohibitiveCPU time. Indeed, the resulting approximated variance of a er2-variance random process is a_Ic --=cr2/nwhere n is the number of independent samples. Despite some techniques which improve the convergence