Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a low-dimensional approximation (or sparse approximation) for some quantity of interest in a model, where just a few orthogonal basis polynomials are required. We consider linear dynamical systems consisting of ordinary differential equations with random variables. The aim of this paper is to explore methods for producing low-dimensional approximations of the quantity of interest further. We investigate two numerical techniques to compute a low-dimensional representation, which both fit the approximation to a set of samples in the time domain. On the one hand, a frequency domain analysis of a stochastic Galerkin system yields the selection of the basis polynomials. It follows a linear least squares problem. On the other hand, a sparse minimization yields the choice of the basis polynomials by information from the time domain only. An orthogonal matching pursuit produces an approximate solution of the minimization problem. We compare the two approaches using a test example from a mechanical application.

中文翻译：

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随机线性动力系统中基选择的时域和频域方法

多项式混沌方法已广泛用于分析不确定性量化中的系统。此外，存在多种方法来确定模型中某些感兴趣量的低维近似（或稀疏近似），其中只需要几个正交基多项式。我们考虑由具有随机变量的常微分方程组成的线性动力系统。本文的目的是进一步探索生成感兴趣数量的低维近似值的方法。我们研究了两种数值技术来计算低维表示，它们都将近似值拟合到时域中的一组样本。一方面，随机伽辽金系统的频域分析产生了基多项式的选择。它遵循线性最小二乘问题。另一方面，稀疏最小化仅通过来自时域的信息产生基多项式的选择。正交匹配追踪产生最小化问题的近似解。我们使用来自机械应用的测试示例来比较这两种方法。