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In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimedthat, regarding Fourier spectral methods for solving differential equations,``[t]he virtues of the Fast Fourier Transform will continue to improve as therelentless march to larger and larger [bandwidths] continues''. This paperattempts to further the virtue of the Fast Fourier Transform (FFT) as not onlybandwidth is pushed to its limits, but also the dimension of the problem.Instead of using the traditional FFT however, we make a key substitution: ahigh-dimensional, sparse Fourier transform (SFT) paired with randomized rank-1lattice methods. The resulting sparse spectral method rapidly and automaticallydetermines a set of Fourier basis functions whose span is guaranteed to containan accurate approximation of the solution of a given elliptic PDE. This muchsmaller, near-optimal Fourier basis is then used to efficiently solve the givenPDE in a runtime which only depends on the PDE's data compressibility andellipticity properties, while breaking the curse of dimensionality andrelieving linear dependence on any multiscale structure in the originalproblem. Theoretical performance of the method is established herein withconvergence analysis in the Sobolev norm for a general class of non-constantdiffusion equations, as well as pointers to technical extensions of theconvergence analysis to more general advection-diffusion-reaction equations.Numerical experiments demonstrate good empirical performance on severalmultiscale and high-dimensional example problems, further showcasing thepromise of the proposed methods in practice.

Sparse Spectral Methods for Solving High-Dimensional and Multiscale Elliptic PDEs