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.

翻译：我们开发了一个高效的、非侵入性的适应性算法,以解决具有随机系数的椭圆部分差异方程式。 稀少的快速傅里叶变换器检测到某个搜索域中最重要的频率, 因此适应性地生成一个适合该函数中大约最大的Fourier系数的适合的Fourier基数。 我们的统一的SFFT同时对空间域中每个有限元素网点的节点同时使用 w.r.t., 并且通过加入检测到的频率集,为所有空间节点创造合适的近似空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间。 这个策略允许更快和更有效地计算, 而不是仅仅对每个节点分别使用完整的 SFFT 算法。 然后我们用周期性、 亲吻性和 逻辑性随机系数测试不同的示例。 结果比使用给定标准频率集时要好得多, 而算法不需要任何关于解决方案的事先信息。