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Determine the point source of the heat equation with sparse boundary measurements

Qiling Gu, Wenlong Zhang, Zhidong Zhang

TL;DR

This work studies the inverse problem of locating a Dirac point source for the heat equation on the unit disk using sparse boundary flux measurements. It combines a spectral-harmonic framework with a least-squares reconstruction controlled by a gradient-descent solver in polar coordinates, and proves a uniqueness result based on two boundary observation points with a diophantine separation condition θ_1−θ_2∉πℚ. The authors develop a forward solver and an inverse solver, and validate the approach through numerical experiments that show accurate source localization and robustness to noise, even with sparse boundary data. The methodology advances point-source localization in diffusion processes by enabling reliable reconstruction from minimal boundary information.

Abstract

In this work the authors consider the recovery of the point source in the heat equation. The used data is the sparse boundary measurements. The uniqueness theorem of the inverse problem is given. After that, the numerical reconstruction is considered. We propose a numerical method to reconstruct the location of a Dirac point source by reformulating the inverse problem as a least-squares optimization problem, which is efficiently solved using a gradient descent algorithm. Numerical experiments confirm the accuracy of the proposed method and demonstrate its robustness to noise.

Determine the point source of the heat equation with sparse boundary measurements

TL;DR

This work studies the inverse problem of locating a Dirac point source for the heat equation on the unit disk using sparse boundary flux measurements. It combines a spectral-harmonic framework with a least-squares reconstruction controlled by a gradient-descent solver in polar coordinates, and proves a uniqueness result based on two boundary observation points with a diophantine separation condition θ_1−θ_2∉πℚ. The authors develop a forward solver and an inverse solver, and validate the approach through numerical experiments that show accurate source localization and robustness to noise, even with sparse boundary data. The methodology advances point-source localization in diffusion processes by enabling reliable reconstruction from minimal boundary information.

Abstract

In this work the authors consider the recovery of the point source in the heat equation. The used data is the sparse boundary measurements. The uniqueness theorem of the inverse problem is given. After that, the numerical reconstruction is considered. We propose a numerical method to reconstruct the location of a Dirac point source by reformulating the inverse problem as a least-squares optimization problem, which is efficiently solved using a gradient descent algorithm. Numerical experiments confirm the accuracy of the proposed method and demonstrate its robustness to noise.

Paper Structure

This paper contains 20 sections, 8 theorems, 78 equations, 10 figures, 9 tables.

Table of Contents

  1. Introduction.
  2. Mathematical statement.
  3. Background and literature.
  4. Main result and outline.
  5. Preliminaries.
  6. Dirichlet eigensystem of -Delta
  7. The set of harmonic functions xi l for l from -infinity to infinity
  8. The proof of Theorem \ref{['thm-unique']}.
  9. Representation of the boundary flux data.
  10. Auxiliary results.
  11. Proof of Theorem \ref{['thm-unique']}.
  12. Numerical algorithms.
  13. Forward solver.
  14. Inverse solver.
  15. Numerical experiments.
  16. ...and 5 more sections

Key Result

Theorem 1.1

Set $z_\ell=(\cos\theta_\ell,\sin\theta_\ell)\in \partial\Omega,\ \ell=1,2$ to be the boundary observation points and suppose the following condition is fulfilled, Given two unknown point source $(r_*,\theta_*)$ and $(\tilde{r}_*,\tilde{\theta}_*)$, we denote corresponding solutions of equation (eq-ibvp) by $u$ and $\tilde{u}$, respectively. If then

Figures (10)

  • Figure 1: The numerical solution $u_h$ of forward problem with $m=n=70$.
  • Figure 2: Observation, Actual and Estimated Points (\ref{['46']}).
  • Figure 3: Observation, Actual and Estimated Points (\ref{['46']}).
  • Figure 4: Observation, Actual and Estimated Points (\ref{['46']}).
  • Figure 5: Observation, Actual and Estimated Points (\ref{['51']}).
  • ...and 5 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more