The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occurring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high-dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance.

中文翻译：

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具有随机系数的ODE的稀疏FFT方法

本文提出了一种求解常微分方程（ODE）的通用策略，其中一些系数取决于空间变量和附加随机变量。该方法基于最近开发的尺寸增量稀疏快速傅里叶变换的应用。由于此类算法需要周期性的信号，因此我们在发生的求解步骤中讨论了周期化策略和相关的必要的去周期化修改。ODE的计算近似解取决于空间变量以及随机变量。当然，高维近似过程的关键挑战之一是评估每个变量对解的影响以及确定变量集内的关系和耦合的速率。