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PDE propagation, sampling, and the Fourier ratio

A. Iosevich, J. Iosevich, E. Palsson, A. Yavicoli

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.

PDE propagation, sampling, and the Fourier ratio

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 which quantifies effective spectral dimension and governs stable 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 . 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 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.
Paper Structure (27 sections, 20 theorems, 149 equations, 4 figures)

This paper contains 27 sections, 20 theorems, 149 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Wave equation motivation
  3. Heat equation motivation
  4. Outlook: Averaging and dynamical smoothing
  5. Organization and main results
  6. Imaging viewpoint: an inverse problem with a PDE-preconditioned prior
  7. Notation and discrete Fourier transform
  8. Main theorems
  9. A priori Fourier ratio bounds for discretized initial data
  10. A Fourier ratio improvement for fixed-time wave snapshots
  11. A Fourier ratio improvement for fixed-time heat snapshots
  12. Time-dependent sampling budgets under sensor loss
  13. Sampling and recovery from missing spatial samples
  14. Reconstruction algorithm and practical implementation
  15. Optimization problem
  16. ...and 12 more sections

Key Result

Proposition 4.1

Let $d\ge 1$ and let $f$ be a real-valued function on $[0,1]^d$ which is $1$-periodic in each variable and belongs to $C^2([0,1]^d)$. Let $g:{\mathbb Z}_N^d\to{\mathbb R}$ be its discretization, Assume that Then there exists a constant $C>0$ depending only on $d$ such that where and

Figures (4)

  • Figure 1: Fourier ratio $FR(g_t)$ for heat snapshots as a function of time $t$ at fixed grid size $N$. The decay illustrates the reduction in spectral complexity induced by diffusion.
  • Figure 2: Empirical recovery success probability as a function of sample size $M$ for heat snapshots.
  • Figure 3: Relative reconstruction error as a function of sample size $M$ for heat snapshots.
  • Figure 4: Fourier ratio $FR(g)$ and $FR(g_t)$ for wave snapshots in dimension three as a function of grid size $N$.

Theorems & Definitions (50)

  • Definition 3.1: Wrapped Euclidean magnitude
  • Proposition 4.1: Fourier ratio bound for discretized initial data
  • Remark 4.2
  • Remark 4.3
  • Definition 4.4: Discrete wave snapshot
  • Remark 4.5
  • Remark 4.6
  • Theorem 4.7: Fourier ratio bound for wave snapshots on ${\mathbb Z}_N^3$
  • Remark 4.8
  • Remark 4.9: Quantitative comparison for the wave equation
  • ...and 40 more