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
β1 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.Comment: 33 pages, 9 figure