Multi-element stochastic spectral projection for high quantile estimation
Introduction
The estimation of high quantile is a challenging numerical topic and has received a considerable amount of attention in many research disciplines [1]. Let us consider a numerical model of a complex system where is a N-tuple vector and is a deterministic numerical model. The numerical model is expensive in terms of the computation time required for a single run. We are interested in the case where its input is a random vector ; in this context, the output is a random scalar. The α-quantile of Y is the level such that the probability that Y takes a value lower than is α:where is the cumulative distribution function (cdf) of Y, i.e. . To accurately estimate high quantiles with the Monte Carlo (MC) method, a large number of samples is required. When the numerical models are computationally expensive to evaluate, the large MC samples required to accurately estimate high quantiles may render this approach impractical.
Different approaches have been developed to improve the accuracy of α-quantile estimation when the number of evaluations on the complete model is limited. As we assume no prior knowledge of the function in question, the estimation of high quantiles usually requires the following steps. First, is roughly estimated. Second, design points, which are the most likely inputs satisfying , are sought. Third, sampling refinements are performed near the design points. High computational cost arises especially in the first step due to the accuracy required in the preliminary quantile estimation and in the second step due to the multiple function evaluations to estimate values, gradients, and Hessians of the complete model. Importance sampling (IS) is a refined Monte Carlo sampling strategy that can be used in the third step to concentrate MC inputs near ; we will compare this strategy with our metamodel-based approach. Note also that, within the context of structural reliability, First-Order Reliability Method (FORM) and Second-Order Reliability Method (SORM) have been developed to estimate quantiles but their errors are difficult to estimate [2], [3], [4].
In the current study, we investigate quantile estimation by multi-element generalized Polynomial Chaos (gPC) metamodels. The gPC has recently been applied to uncertainty quantification studies [5], [6], [7]. By approximating the complete model with an accurate spectral metamodel, different statistical measures such as mean, variance and Sobol’ sensitivity indices can be readily computed [7]. Large number of MC samples can be rapidly evaluated on the metamodel to estimate the quantile of the model solution. In addition, the metamodel can be used in the design point search algorithm to estimate the values of the inputs corresponding to the solution quantiles and the metamodel approach is valid even for complex nonlinear complete models with multiple design points. In comparison to the multiple design point algorithm proposed in [8], the current approach based on gPC metamodels do not require numerical modifications to in order to locate all design points.
As the gPC metamodels are spectral expansions about the means of the inputs, their accuracy may worsen away from these mean values where the extreme events may occur. By increasing the global approximation level, we may eventually increase the accuracy of the global metamodel away from the mean but it can be very expensive. Thus, a multi-element approach is used by combining the global metamodel with supplementary local metamodels centered at the design points. Similar multi-element approaches have been used previously although the local refinement goal was to reduce the contribution to the global variance from each local element [9], [10], [11], [12].
The current approach is similar to the parametric study of the random inputs in [13], [14] with the exception that the complete models are replaced by their metamodels. Indeed metamodels have been used previously to improve the cost of quantile estimation. Bucher & Bourgund have used metamodels constructed from polynomial expansions without cross terms [15] and polynomial chaos metamodels were used in [16]. Despite the availability of the metamodel, these studies used conventional gradient descent algorithms to determine the design points, which increases the sampling costs significantly. Moreover, polynomials with infinite supports are used to construct localized refinements around the design points [16]; the accuracy of the expansion can be improved if appropriate polynomials with finite supports are used.
The methodology used in the current study is outlined in Fig. 1. Quantile estimation methods are described in Section 2. The construction of the gPC metamodel is detailed in Section 3. Once the preliminary estimation of the α-quantile of Y is obtained from the global gPC metamodel, the Lagrange multiplier method is used to search for the design points and the details applied to the metamodels of this approach are outlined in Section 4. Section 5 illustrates how the local gPC metamodels in the multi-element approach are combined with the global metamodel to estimate the α-quantile. Finally, the proposed methods are applied to some examples in Section 6.
Section snippets
Quantile estimation
The current state of art for quantile estimation in the context of failure probability is documented in details in [17]. Key methods and their application to quantile estimation are presented in the following section.
Generalized polynomial chaos
The numerical model from which quantiles are evaluated can often be complex and computationally expensive. The computation cost can be significantly reduced by the use of metamodels. Metamodels can be either a reduced numerical code of the complete model or a response surface calibrated to mimic the complete model for typical realizations of the input parameters. The latter is the focus of this Section. In the context of quantile estimation, supplementary metamodels are constructed around
Design point search
In the parameter space, the feasibility surface of the complete model is the surface . From now on, we assume the components of the random input vector are independent and identically distributed random variables with Gaussian distribution, zero mean and unit variance. For dependent random vectors not in the standard normal space, the Rosenblatt or Nataf bijective transformations can be used to transform them into the desired standard normal space [40], [41]. For independent
Local refinement
Once the design points are determined from the global metamodel, the function response in their neighborhoods can be refined. The following sections describe the two steps of this refinement procedure, namely the definition of the local refinement domain for the ith design point and the construction of the multi-element gPC metamodel.
Examples
A Gaussian-like and a hyperbolic tangent functions are used to validate the MEgPC approach in the next two sections. First, the accuracy in the design points determined by GPM and gPC LMM are first compared at selected random dimension N. Second, the convergences of errors in the empirical α-quantile estimated with different numerical methods are demonstrated at selected N. Finally, the different quantile estimation methods are compared at different N values. After the MEgPC approach is
Conclusion
A multi-element gPC (MEgPC) metamodel is used to estimate high quantiles of solutions of multivariate numerical models which are computationally expensive to evaluate. The multi-element metamodel consists of a global gPC metamodel in the standard normal domain and local gPC metamodels in bounded domains centered at design points corresponding to the quantile. The metamodels are constructed using a non-intrusive method where the numerical solver is sampled on fixed and deterministic quadrature
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