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Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements

Fangyu Gong, Bangti Jin, Yavar Kian, Sizhe Liu

Abstract

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in $\mathbb{R}^d$ ($d\geq2$), and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth and bounded in $\mathbb{R}^2$, under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.

Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements

Abstract

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in (), and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth and bounded in , under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.
Paper Structure (9 sections, 10 theorems, 100 equations, 2 figures, 4 tables)

This paper contains 9 sections, 10 theorems, 100 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on the direct problem
  3. Regularity and extension of the solution
  4. Heat kernel, Green's function and Poisson kernel
  5. Dirichlet eigenfunctions on the unit ball
  6. Proof of the main results
  7. Proof of Theorem \ref{['thm:ball']}
  8. Proof of Theorem \ref{['thm:compact']}
  9. Numerical experiments and discussions

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^d$ be the unit ball with $d\geq 2$, and $g,\widetilde{g}\in \mathcal{A}_{\rm pwc}$. Let $u$ and $\widetilde{u}$ solve problem eqn:heat with the source $F=g(t)\delta(x-p)$ and $\widetilde{F}=\widetilde{g}(t)\delta(x-\widetilde{p})$, respectively. Suppose that $\partial_{

Figures (2)

  • Figure 1: The flux traces and recovered location for Example \ref{['exam:pwc-g']} with $\delta=1\%$ (top), $\delta =3\%$ (middle) and $\delta=5\%$ (bottom).
  • Figure 2: The numerical results for Example \ref{['exam:cpt-g']} with $\delta=1\%$ (top), $\delta=3\%$ (middle) and $\delta=5\%$ (bottom).

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Lemma 2.1: Grebenkov2013
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 12 more