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

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-04-02 DOI:10.1007/s10208-024-09649-8

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Abstract

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.

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求解高维和多尺度椭圆 PDE 的稀疏谱方法

摘要 John Boyd 在其专著《Chebyshev 和傅立叶频谱方法》中声称，关于求解微分方程的傅立叶频谱方法，"随着向更大[带宽]的无情进军，快速傅立叶变换的优点将不断改进"[Boyd in Chebyshev and Fourier spectral methods, second rev ed., Dover Publications, Mineola NY, 2001, pg. 194]。Dover Publications, Mineola, NY, 2001, pg. 194]。本文试图进一步发挥快速傅立叶变换 (FFT) 的优势，因为它不仅将带宽推向了极限，还将问题的维度推向了极限。然而，我们并没有使用传统的 FFT，而是做了一个关键的替换：将高维稀疏傅立叶变换与随机秩-1 网格方法结合起来。由此产生的稀疏谱方法能快速自动地确定一组傅里叶基函数，其跨度保证包含给定椭圆 PDE 解的精确近似值。然后，利用这个更小的、接近最优的傅立叶基函数来高效求解给定的 PDE，其运行时间仅取决于 PDE 的数据可压缩性和椭圆特性，同时打破了维度诅咒，并解除了对原始问题中任何多尺度结构的线性依赖。本文通过对一般类别的非恒定扩散方程进行索博列夫规范收敛分析，确定了该方法的理论性能，并指出了将收敛分析扩展到更一般的平流-扩散-反应方程的技术途径。数值实验在几个多尺度和高维示例问题上证明了良好的经验性能，进一步展示了所提方法在实践中的前景。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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