随机系数ODE的稀疏FFT方法
A sparse FFT approach for ODE with random coefficients
摘要：
本文给出了求解某些系数依赖于空间变量和附加随机变量的常微分方程的一般策略。该方法是基于最近开发的维数增量稀疏快速傅里叶变换的应用。由于这样的算法需要周期性的信号，我们讨论周期化策略和相关的必要deperiodization修改内发生的解决方案的步骤。常微分方程的计算近似解依赖于空间变量，也依赖于随机变量。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。所建议的方法以全自动的方式以合理的计算成本满足这些挑战，即，与现有的方法相比，不需要预先严格限制所使用的Analog函数集。
英文摘要：
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 occuring 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.
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