The uniform sparse FFT with application to PDEs with random coefficients

TL;DR

This paper introduces the uniform sparse FFT (usFFT), an adaptive algorithm for efficiently solving elliptic PDEs with random coefficients by detecting key frequencies across multiple spatial nodes simultaneously.

Contribution

The paper presents the usFFT, a novel non-intrusive, adaptive Fourier-based method that efficiently handles multiple spatial nodes in PDEs with random coefficients.

Findings

usFFT reduces the number of samples needed compared to traditional methods

The algorithm effectively handles different types of random coefficients

Numerical tests demonstrate improved efficiency and accuracy

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.