Polynomial chaos expansions for dependent random variables
Polynomial chaos expansions for dependent random variables
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DOI：
10.1016/j.cma.2019.03.049
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发表时间：
2019-03
影响因子：
7.2
通讯作者：
J. Jakeman;F. Franzelin;A. Narayan;M. Eldred;Dirk Plfueger
中科院分区：
文献类型：
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作者：
J. Jakeman;F. Franzelin;A. Narayan;M. Eldred;Dirk Plfueger
多项式混沌展开（PCE）非常适合于量化由独立随机变量参数化的模型中的不确定性。独立的假设导致简单的策略，建立多元标准正交基和抽样策略，以评估PCE系数。相比之下，PCE应用于因变量模型更具挑战性。三种方法可用于构建因变量模型的PCE。第一种方法使用映射方法，其中测量变换，如Nataf和Rosenblatt变换，可以用来映射依赖的随机变量到独立的，但是我们表明，这可能会显着降低性能，因为映射的雅可比矩阵必须近似。第二种策略是支配支持方法类。在这些方法中，PCE是使用独立的随机变量，其分布支持占主导地位的支持真正的依赖联合密度，我们提供的证据表明，这种方法似乎产生近似次优精度。第三种方法，这里提出的新方法，使用Gram-Schmidt正交化（GSO）数值计算正交多项式的相关随机变量。这种方法已被成功地使用时，使用侵入式随机Galerkin方法求解微分方程，在本文中，我们使用GSO建立PCE使用非侵入式随机配置方法。随机配置方法将模型视为黑箱，并从一组样本中构建输入-输出映射的近似值。从样本构建PCE可能会引入病态，但不会困扰随机Galerkin方法。为了减轻这种病态，我们生成加权Leja序列，这是嵌套的样本集，以建立准确的多项式插值。我们表明，我们提出的方法，GSO加权Leja序列，产生PCE的数量级更准确的PCE使用映射或主导支持方法构建。
Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram–Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input–output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.