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，其应用范围从金融学延伸至计算化学。然而，求解这类方程的标准数值方法通常受制于维度诅咒。本文聚焦于定义在高维区域且具有周期边界条件的稳态扩散方程，致力于克服这一挑战。受近期高维稀疏函数逼近领域进展的启发，我们提出了一种名为"压缩傅里叶配点法"的新方法。该方法融合了压缩感知与谱配点法的思想，用蒙特卡洛采样替代结构化配点网格，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。我们进行了严格的理论分析，证明所提方法的逼近误差与解的最佳$s$项（基于傅里叶基）逼近误差相当。利用近期提出的有界Riesz系统随机采样框架，我们的分析表明：在扩散系数正则性满足充分条件的前提下，压缩傅里叶配点法在配点数量维度上缓解了维度诅咒。最后通过数值实验展示了该方法在逼近稀疏解和可压缩解时的精度与稳定性。