We consider the effect of multiple stochastic parameters on the time-average quantities of chaotic systems. We employ the recently proposed sensitivity-enhanced generalized polynomial chaos expansion, se-gPC, to quantify efficiently this effect. se-gPC is an extension of gPC expansion, enriched with the sensitivity of the time-averaged quantities with respect to the stochastic variables. To compute these sensitivities, the adjoint of the shadowing operator is derived in the frequency domain. Coupling the adjoint operator with gPC provides an efficient uncertainty quantification algorithm, which, in its simplest form, has computational cost that is independent of the number of random variables. The method is applied to the Kuramoto-Sivashinsky equation and is found to produce results that match very well with Monte Carlo simulations. The efficiency of the proposed method significantly outperforms sparse-grid approaches, such as Smolyak quadrature. These properties make the method suitable for application to other dynamical systems with many stochastic parameters.

中文翻译：

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使用灵敏度增强多项式混沌展开对混沌系统的时间平均量进行不确定性量化

我们考虑多个随机参数对混沌系统时间平均量的影响。我们采用最近提出的灵敏度增强广义多项式混沌展开（se-gPC）来有效量化这种效应。 se-gPC 是 gPC 扩展的扩展，增强了时间平均量对随机变量的敏感性。为了计算这些灵敏度，在频域中导出阴影算子的伴随。将伴随算子与 gPC 耦合提供了一种有效的不确定性量化算法，其最简单的形式具有与随机变量数量无关的计算成本。该方法应用于 Kuramoto-Sivashinsky 方程，并发现产生的结果与蒙特卡洛模拟非常匹配。该方法的效率明显优于稀疏网格方法，例如 Smolyak 求积法。这些特性使得该方法适合应用于具有许多随机参数的其他动力系统。