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：高维偏微分方程是重要的数学模型工具，应用领域涵盖金融计算与计算化学等。然而，求解此类方程的标准数值方法通常受制于维数灾难。本文聚焦于高维周期边界条件下稳态扩散方程的求解问题。受近期高维稀疏函数逼近研究进展的启发，我们提出一种名为“压缩傅里叶配点法”的新方法。该方法融合压缩感知与谱配点法的思想，采用蒙特卡洛采样替代结构化配点网格，并运用正交匹配追踪和$\ell^1$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。我们进行了严格的理论分析，证明所提方法的逼近误差可与解的最优$s$-项傅里叶逼近误差相媲美。基于有界Riesz系统随机采样的最新理论框架，分析表明：在扩散系数正则性满足充分条件时，该压缩傅里叶配点法可在配点数量层面缓解维数灾难。此外，数值实验验证了该方法对稀疏解与可压缩解逼近的精确性与稳定性。