A Probabilistic Radial Basis Function Approach for Uncertainty Quantification

Interest in uncertainty quantification is rapidly increasing, since inherent physical variations cannot be neglected in computational modeling with increasing accuracy of the deterministic computations. In Computational Fluid Dynamics (CFD) many uncertainties can be present, such as flow parameters, boundary conditions, and the geometry. The amount of work, however, increases significantly when uncertainty quantification is applied. Since deterministic CFD can already be computationally intensive, an efficient uncertainty quantification method is required. When the number of uncertain parameters becomes large, a high dimensional response surface has to be computed. In this paper, radial basis functions (RBFs) are used since they are known to be efficient interpolants in high dimensional spaces. This Probabilistic Radial Basis Function (PRBF) approach is applied to several test problems with multiple uncertain parameters to gain some first insights. Combinations of different radial basis functions and sampling techniques are used to study the performance of different combinations. First the PRBF approach is applied to a mass-spring problem, for which the convergence of the RBFs and the influence of the sampling techniques is investigated. The second test case is the piston problem with an external forcing, for which the number of uncertain parameters is increased. The results of the PRBF approach are compared with those obtained from the Probabilistic Collocation method. Finally, the method is applied to an airfoil with uncertain free stream conditions and geometry. The flow is computed for a limited number of deterministic turbulent Navier-Stokes computations using a commercial CFD solver. 1.0 INTRODUCTION Advanced algorithms and increasing computer power lead to accurate deterministic simulations. The physical systems that are modeled, often have variability which is neglected. There is interest in modelling this variability in complex systems, since it can influence the solution significantly. Uncertainty quantification is used to compute the probability distribution of the solution based on uncertain input parameters. When uncertainty quantification is used in combination with existing flow solvers ideally the uncertainty quantification method is non-intrusive. A non-intrusive method requires several deterministic solves using the deterministic solver as a black-box. Efficient non-intrusive methods are for example the Probabilistic Collocation method [1, 10, 11, 16] and the Non-intrusive Polynomial Chaos method [8, 9], which are both based on the Polynomial Chaos method [7]. For multiple uncertain parameters the amount of deterministic computations grows rapidly. For the Probabilistic Collocation method the number of points is equal to (p + 1)d, with p the order of the approximation and d the number of uncertain parameters. As an alternative sparse grid approaches [5, 6, 17] can be used to increase the RTO-MP-AVT-147 NATO/PFP UNCLASSIFIED 35 1 A Probabilistic Radial Basis Function Approach for Uncertainty Quantification efficiency. For the Non-intrusive Polynomial Chaos method the number of coefficients is (d + p)!/d!p!. Hosder, Walters, and Balch [8] showed that for a good Non-intrusive Polynomial Chaos approximation the amount of sampling points should be twice the number of coefficients. The polynomial chaos based methods use a global polynomial approximation of the response surface. In this paper the response surface is approximated using Radial Basis Functions (RBFs) [2] through a limited number of support points. RBFs are used since they are known to be efficient interpolants in high dimensional spaces. The support points can be chosen by several sampling strategies. Here several combinations of different RBFs and sampling techniques are investigated to gain some first insight in the use of RBFs for response surface approximation. Recently, RBFs [2, 12, 14] became more popular for response surface approximation. Regis and Shoemaker [14] proposed a stochastic Radial Basis Function method for global optimization problems of expensive functions. They define expensive functions as function which take from minutes to several hours to evaluate. Here the focus is on CFD, where simulations take from hours to days or even weeks to compute, so the number of available support points is minimal. The PRBF approach is applied to three test cases. The first test case is the mass-spring problem, with the spring stiffness and mass uncertain. A comparison of several commonly used RBFs and sampling techniques is made based on the convergence of the mean and variance with respect to the number of samples. Secondly, the piston problem with a forcing boundary condition is employed with four uncertain parameters. The results are compared to those obtained from the Probabilistic Collocation method. The third test case is a turbulent Navier-Stokes computations around a NACA0012 airfoil with four uncertain parameters. The free stream Mach number and angle of attack are assumed to be uncertain, as well as the geometry of the airfoil. The NACA0012 airfoil is parametrized by the maximum camber, maximum camber location and relative thickness. Here the maximum camber and relative thickness are assumed to be uncertain. This paper is organized as follows; first the PRBF approach is explained in section 2.0. Section 3.0 evaluates the RBFs and sampling methods using the mass-spring problem in section 3.1 and the piston problem in section 3.2. Next section 4.0 shows the application of the PRBF approach for a turbulent Navier-Stokes computation around a NACA0012 airfoil with four uncertain parameters. Section 5.0 provides the conclusions and the final section 6.0 gives input for further research. 2.0 PROBABILISTIC RADIAL BASIS FUNCTION APPROACH This section introduces the RBF framework and some commonly used functions. Furthermore, the sampling techniques used to obtain the support points for the RBFs are discussed. 2.1 Radial Basis Functions Consider a problem with d uncertain parameters a1(ω), a2(ω), . . . , ad(ω). The randomness of the parameters is indicated by ω ∈ Ω, which is a random event from the set of outcomes Ω. The probability space is given by (Ω,F , P ), with F ⊂ 2Ω the σ-algebra of events and P a probability measure. The parameter space is a d-dimensional probability space a(ω) = {a1(ω), a2(ω), . . . , ad(ω)}. The response surface u(a(ω)) is approximated by