Elsevier

Journal of Computational Physics

Volume 243, 15 June 2013, Pages 87-108
Journal of Computational Physics

Multi-element stochastic spectral projection for high quantile estimation

https://doi.org/10.1016/j.jcp.2013.01.012Get rights and content

Abstract

We investigate quantile estimation by multi-element generalized Polynomial Chaos (gPC) metamodel where the exact numerical model is approximated by complementary metamodels in overlapping domains that mimic the model’s exact response. The gPC metamodel is constructed by the non-intrusive stochastic spectral projection approach and function evaluation on the gPC metamodel can be considered as essentially free. Thus, large number of Monte Carlo samples from the metamodel can be used to estimate α-quantile, for moderate values of α. As the gPC metamodel is an expansion about the means of the inputs, its accuracy may worsen away from these mean values where the extreme events may occur. By increasing the approximation accuracy of the metamodel, we may eventually improve accuracy of quantile estimation but it is very expensive. A multi-element approach is therefore proposed by combining a global metamodel in the standard normal space with supplementary local metamodels constructed in bounded domains about the design points corresponding to the extreme events. To improve the accuracy and to minimize the sampling cost, sparse-tensor and anisotropic-tensor quadratures are tested in addition to the full-tensor Gauss quadrature in the construction of local metamodels; different bounds of the gPC expansion are also examined. The global and local metamodels are combined in the multi-element gPC (MEgPC) approach and it is shown that MEgPC can be more accurate than Monte Carlo or importance sampling methods for high quantile estimations for input dimensions roughly below N=8, a limit that is very much case- and α-dependent.

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 y=f(x) where x is a N-tuple vector and f:RNR 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 X=(X1,,XN); in this context, the output Y=f(X) is a random scalar. The α-quantile of Y is the level yα such that the probability that Y takes a value lower than yα is α:yα=inf{y;F(y)α},where F(y) is the cumulative distribution function (cdf) of Y, i.e. F(y)=P(Yy). 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, yα is roughly estimated. Second, design points, which are the most likely inputs ξα satisfying f(ξα)=yα, 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 ξα; 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 f(x) 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 {xRN,f(x)-yα=0}. 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 ξˆαi,r 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 Dβi for the ith design point ξˆαi,r 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

References (48)

  • R. Ghanem et al.

    Stochastic Finite Elements: A Spectral Approach

    (1991)
  • D. Xiu et al.

    Stochastic modeling of flow-structure interactions using generalized polynomial chaos

    Journal of Fluids Engineering

    (2001)
  • J. Ko et al.

    Sensitivity of two-dimensional spatially developing mixing layers with respect to uncertain inflow conditions

    Physics of Fluids

    (2008)
  • X. Wang et al.

    An adaptive multi-element generalized polynomial chaos method for stochastic differential equations

    Journal of Computational Physics

    (2005)
  • X. Wan et al.

    Multi-element generalized polynomial chaos for arbitrary probability measures

    SIAM Journal of Scientific Computing

    (2006)
  • J.L. Meitour et al.

    Prediction of stochastic limit cycle oscillations using an adaptive polynomial chaos method

    Journal of Aeroelasticity and Structural Dynamics

    (2010)
  • M. Barbato, Finite element response sensitivity, probabilitic response and reliability analyses of structrual systems...
  • Q. Gu, Finite element response sensitivity and reliability analyses of soil-foundation structure-interaction systems,...
  • M.M. Zuniga, Méthodes stochastiques pour l’estimation contrôlée de faibles probabilités sur des modèles physiques...
  • G.S. Fishman, Monte Carlo. Concepts, algorithms, and applications, Springer, New York,...
  • H.A. David et al.

    Order Statistics

    (2003)
  • S. Wilks

    Mathematical Statistics

    (1962)
  • R.R. Bahadur

    A note on quantiles in large samples

    The Annals of Mathematical Statistics

    (1966)
  • P.W. Glynn, Importance sampling for monte carlo estimation of quantiles, in: Proceedings of the Second International...
  • Cited by (2)

    • The algebraic method in quadrature for uncertainty quantification

      2016, SIAM-ASA Journal on Uncertainty Quantification
    View full text