In the context of uncertainty quantification, computational models are required to be repeatedly evaluated. This task is intractable for costly numerical models. Such a problem turns out to be even more severe for stochastic simulators, the output of which is a random variable for a given set of input parameters. To alleviate the computational burden, surrogate models are usually constructed and evaluated instead. However, due to the random nature of the model response, classical surrogate models cannot be applied directly to the emulation of stochastic simulators. To efficiently represent the probability distribution of the model output for any given input values, we develop a new stochastic surrogate model called stochastic polynomial chaos expansions. To this aim, we introduce a latent variable and an additional noise variable, on top of the well-defined input variables, to reproduce the stochasticity. As a result, for a given set of input parameters, the model output is given by a function of the latent variable with an additive noise, thus a random variable. Because the latent variable is purely artificial and does not have physical meanings, conventional methods (pseudo-spectral projections, collocation, regression, etc.) cannot be used to build such a model. In this paper, we propose an adaptive algorithm that does not require repeated runs of the simulator for the same input parameters. The performance of the proposed method is compared to the generalized lambda model and a state-of-the-art kernel estimator on two case studies in mathematical finance and epidemiology and on an analytical example whose response distribution is bimodal. The results show that the proposed method is able to accurately represent general response distributions, i.e., not only normal or unimodal ones. In terms of accuracy, it generally outperforms both the generalized lambda model and the kernel density estimator.

中文翻译：

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模拟随机模拟器的随机多项式混沌展开

在不确定性量化的背景下，需要对计算模型进行反复评估。对于昂贵的数值模型来说，这项任务是棘手的。对于随机模拟器来说，这样的问题更为严重，对于一组给定的输入参数，其输出是一个随机变量。为了减轻计算负担，通常会构建和评估替代模型。然而，由于模型响应的随机性，经典替代模型不能直接应用于随机模拟器的仿真。为了有效地表示任何给定输入值的模型输出的概率分布，我们开发了一种称为随机多项式混沌展开的新随机代理模型。为此，我们引入了一个潜在变量和一个额外的噪声变量，在定义明确的输入变量之上，以重现随机性。结果，对于一组给定的输入参数，模型输出由具有附加噪声的潜在变量的函数给出，因此是随机变量。由于潜变量是纯人工的，不具有物理意义，因此无法使用常规方法（伪光谱投影、搭配、回归等）来构建这样的模型。在本文中，我们提出了一种自适应算法，不需要针对相同的输入参数重复运行模拟器。在数学金融学和流行病学的两个案例研究以及响应分布为双峰的分析示例中，将所提出方法的性能与广义 lambda 模型和最先进的核估计器进行了比较。结果表明，所提出的方法能够准确地表示一般响应分布，即，不仅是正常的或单峰的。在准确性方面，它通常优于广义 lambda 模型和核密度估计器。