High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

翻译：高维偏微分方程是一种流行的数学建模工具，其应用范围从金融领域延伸至计算化学。然而，求解这些方程的标准数值技术通常会受到维数灾难的影响。本文针对定义在高维周期边界条件域上的稳态扩散方程，致力于解决这一挑战。受近期高维稀疏函数逼近研究的启发，我们提出了一种名为"压缩傅里叶配置法"的新方法。该方法融合压缩感知与谱配置的思想，用蒙特卡洛采样替代结构化配置网格，并采用正交匹配追踪和$\ell^1$最小化等稀疏恢复技术来逼近偏微分方程解的傅里叶系数。理论分析表明，所提方法的逼近误差与解的（关于傅里叶基的）最优$s$项逼近误差相当。利用近期提出的有界Riesz系统随机采样框架，我们的分析证明：在扩散系数正则性满足充分条件的情况下，压缩傅里叶配置法在配置点数量维度上能有效缓解维数灾难。数值实验进一步验证了该方法在逼近稀疏解和可压缩解时的精度与稳定性。