We develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The sparse Fast Fourier Transform detects the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. Our uniform sFFT does this w.r.t. the stochastic domain simultaneously for every node of a finite element mesh in the spatial domain and creates a suitable approximation space for all spatial nodes by joining the detected frequency sets. This strategy allows for a faster and more efficient computation, than just using the full sFFT algorithm for each node separately. We then test the usFFT for different examples using periodic, affine and lognormal random coefficients. The results are significantly better than when using given standard frequency sets and the algorithm does not require any a priori information about the solution.

中文翻译：

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均匀稀疏 FFT 应用于具有随机系数的偏微分方程

我们开发了一种高效、非侵入式、自适应算法，用于求解具有随机系数的椭圆偏微分方程。稀疏快速傅立叶变换检测给定搜索域中最重要的频率，因此自适应地生成与函数的近似最大傅立叶系数相对应的合适的傅立叶基。我们的统一 sFFT 对空间域中有限元网格的每个节点同时执行随机域，并通过加入检测到的频率集为所有空间节点创建合适的近似空间。这种策略允许更快、更有效的计算，而不是单独对每个节点使用完整的 sFFT 算法。然后，我们使用周期性、仿射和对数正态随机系数测试不同示例的 usFFT。