A compressive spectral collocation method for the diffusion equation under the restricted isometry property

TL;DR

This paper introduces a novel compressive spectral collocation method for solving PDEs, especially the diffusion equation, leveraging the restricted isometry property to reduce computational cost while maintaining accuracy.

Contribution

It develops a new spectral collocation approach based on compressive sensing principles that requires fewer collocation points for sparse solutions, with theoretical guarantees.

Findings

The method satisfies the restricted isometry property under certain conditions.

It significantly reduces computational cost compared to full spectral collocation.

Numerical results confirm maintained accuracy with fewer collocation points.

Abstract

We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at randomized points, by taking advantage of the compressive sensing principle. The proposed approach makes use of a number of collocation points substantially less than the number of basis functions when the solution to recover is sparse or compressible. Focusing on the case of the diffusion equation, we prove that, under suitable assumptions on the diffusion coefficient, the matrix associated with the compressive spectral collocation approach satisfies the restricted isometry property of compressive sensing with high probability. Moreover, we demonstrate the ability of the proposed method to reduce the computational cost associated with the corresponding full spectral collocation approach while preserving good accuracy through numerical illustrations.