PDE propagation, sampling, and the Fourier ratio

TL;DR

This paper investigates how PDE propagation influences spectral properties and improves the stability of reconstructing discretized fields from incomplete samples, providing theoretical bounds and numerical validation.

Contribution

It demonstrates that PDE propagation can enhance Fourier ratio bounds, leading to more efficient stable recovery from incomplete spatial samples.

Findings

PDE propagation improves Fourier ratio bounds compared to initial discretization.

Wave snapshots in 3D introduce high-frequency decay, stabilizing bounds across grid sizes.

Heat equation frequency damping yields bounds independent of grid resolution.

Abstract

We study recovery from incomplete random spatial samples for discretized fields arising as fixed-time snapshots of partial differential equations. The organizing parameter is the Fourier ratio $$ FR(g)=\frac{\|\widehat g\|_1}{\|\widehat g\|_2}, $$ which quantifies effective spectral dimension and governs stable $\ell^1$ recovery in bounded orthonormal sampling models. Our main observation is that fixed-time PDE propagation can strictly improve Fourier ratio bounds relative to the discretized initial data. In dimension three, the wave snapshot operator introduces additional high-frequency decay, leading after discretization to Fourier ratio bounds that are uniformly controlled in the grid size (up to discretization errors), whereas the corresponding bounds for the initial discretization are typically polynomial in $N$. For the heat equation in any dimension, Gaussian frequency damping yields Fourier ratio bounds that are essentially independent of grid resolution for fixed positive time. Combining these deterministic Fourier ratio improvements with standard $\ell^1$ recovery guarantees yields explicit sampling-rate bounds for stable reconstruction from missing spatial samples. Numerical experiments confirm that PDE propagation acts as a spectral preconditioner that lowers effective sampling complexity in practice.