The uniform sparse FFT with application to PDEs with random coefficients

Abstract

We develop the uniform sparse Fast Fourier Transform (usFFT), an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The algorithm is an adaption of the sparse Fast Fourier Transform (sFFT), a dimension-incremental algorithm, which tries to detect 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. The usFFT does this w.r.t. the stochastic domain of the PDE simultaneously for multiple fixed spatial nodes, e.g., nodes of a finite element mesh. The key idea of joining the detected frequency sets in each dimension increment results in a Fourier approximation space, which fits uniformly for all these spatial nodes. This strategy allows for a faster and more efficient computation due to a significantly smaller amount of samples needed, than just using other algorithms, e.g., the sFFT for each spatial node separately. We test the usFFT for different examples using periodic, affine and lognormal random coefficients in the PDE problems