In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, “[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues” [Boyd in Chebyshev and Fourier spectral methods, second rev ed. Dover Publications, Mineola, NY, 2001, pg. 194]. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform paired with randomized rank-1 lattice methods. The resulting sparse spectral method rapidly and automatically determines a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given elliptic PDE. This much smaller, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE’s data compressibility and ellipticity properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Theoretical performance of the method is established herein with convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations, as well as pointers to technical extensions of the convergence analysis to more general advection–diffusion–reaction equations. Numerical experiments demonstrate good empirical performance on several multiscale and high-dimensional example problems, further showcasing the promise of the proposed methods in practice.

约翰·博伊德 (John Boyd)在他的专着《切比雪夫和傅里叶谱方法》中声称，关于求解微分方程的傅里叶谱方法，“随着不断向越来越大的[带宽]迈进，快速傅里叶变换的优点将继续改进。 ” [博伊德在切比雪夫和傅立叶光谱方法中，第二版。多佛出版社，米尼奥拉，纽约，2001 年，第 17 页。 194]。本文试图进一步发挥快速傅里叶变换 (FFT) 的优点，因为不仅带宽被推到了极限，而且问题的维度也被推到了极限。然而，我们没有使用传统的 FFT，而是进行了关键替换：高维稀疏傅里叶变换与随机 1 阶格方法配对。由此产生的稀疏谱方法可以快速、自动地确定一组傅立叶基函数，其跨度保证包含给定椭圆偏微分方程解的精确近似。然后，使用这个小得多、接近最优的傅立叶基在运行时有效地求解给定的偏微分方程，这仅取决于偏微分方程的数据可压缩性和椭圆性属性，同时打破维数诅咒并减轻对原始数据中任何多尺度结构的线性依赖问题。本文通过对一类非恒定扩散方程的一般类 Sobolev 范数进行收敛分析，以及将收敛分析技术扩展到更一般的平流扩散反应方程来建立该方法的理论性能。数值实验在几个多尺度和高维示例问题上展示了良好的经验性能，进一步展示了所提出的方法在实践中的前景。