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Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

Kuang Huang, Bangti Jin, Yavar Kian, Georges Sadaka, Zhi Zhou

TL;DR

This work tackles the inverse problem of identifying point sources, their time-dependent strengths, and the initial state in a subdiffusion model governed by a Caputo time derivative $\partial_t^{\alpha}$ and a general elliptic operator $\mathcal{A}$. It develops a direct-problem theory in the transposition sense and proves improved local regularity, enabling a unique continuation-based identifiability framework that recovers all source parameters from interior measurements on a subdomain near the final time, even with time-dependent coefficients in 1D. Theoretical results cover multi-dimensional domains and extend parabolic-type identifiability to subdiffusion, while numerical experiments using Levenberg–Marquardt confirm feasibility of reconstruction in 1D, 2D, and 3D. The memory effect of subdiffusion is leveraged to perform a posteriori source identification, with practical implications for pollution monitoring and related applications. Overall, the paper advances the theory and practice of inverse source problems for fractional diffusion by establishing well-posedness, uniqueness, and computational strategies for recovering moving point sources.

Abstract

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.

Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

TL;DR

This work tackles the inverse problem of identifying point sources, their time-dependent strengths, and the initial state in a subdiffusion model governed by a Caputo time derivative and a general elliptic operator . It develops a direct-problem theory in the transposition sense and proves improved local regularity, enabling a unique continuation-based identifiability framework that recovers all source parameters from interior measurements on a subdomain near the final time, even with time-dependent coefficients in 1D. Theoretical results cover multi-dimensional domains and extend parabolic-type identifiability to subdiffusion, while numerical experiments using Levenberg–Marquardt confirm feasibility of reconstruction in 1D, 2D, and 3D. The memory effect of subdiffusion is leveraged to perform a posteriori source identification, with practical implications for pollution monitoring and related applications. Overall, the paper advances the theory and practice of inverse source problems for fractional diffusion by establishing well-posedness, uniqueness, and computational strategies for recovering moving point sources.

Abstract

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.

Paper Structure

This paper contains 12 sections, 8 theorems, 80 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Analysis of the direct problem
  3. Solution in the transposition sense
  4. Improved regularity
  5. The proofs of the main theorems
  6. The proof of Theorem \ref{['t1']}
  7. The proof of Theorem \ref{['t1D']}
  8. Numerical results and discussions
  9. 1D numerical examples
  10. 2D numerical examples
  11. 3D numerical examples
  12. Conclusion

Key Result

Theorem 1.1

For $j=1,2$, let $N_j\in\mathbb N$, $\{\lambda_k^j\}_{k=1}^{N_j}\subset H_\alpha(0,T)$ be $N_j$ non-uniformly vanishing functions, $\{x_k^j\}_{k=1}^{N_j}$ be $N_j$ distinct points in $\Omega$ and let $u_0^j\in H^1_0(\Omega)$. Fix also $\omega\subset \Omega\setminus\{x_1^1,\ldots,x_{N_1}^1,x_1^2,\ldo implies and moreover, there exists a one-to-one map $\pi$ of $\{1,\ldots,N\}$ such that

Figures (7)

  • Figure 1: The recovered $\lambda$ and convergence for Example \ref{['exam1']} with $\delta=2\%$ (top) and $\delta=10\%$ (bottom). (a)--(b) and (c)--(d) are for $\epsilon=\frac{3}{4}T$ and $\epsilon=\frac{1}{2}T$, respectively.
  • Figure 2: The recovered $\lambda_i$ and convergence for Example \ref{['exam2']}(i) with $\delta=2\%$ (top) and $\delta=10\%$ (bottom).
  • Figure 3: Numerical results for Example \ref{['exam2']}(ii): the recovered $\lambda_i$ and $u_0$ with $\alpha=0.1$ (top) and the error of the recovered $\lambda_i$ and $u_0$ at different $\alpha$ (bottom).
  • Figure 4: The recovered $\lambda_i$ and convergence for Example \ref{['exam4']}. (a)-(b) and (c)-(d) are for $\epsilon= \frac{3}{4}T$ and $\epsilon=\frac{1}{2}T$, respectively.
  • Figure 5: The recovered $\lambda_i$ and convergence for Example \ref{['exam5']} with $\epsilon=\frac{3}{4}T$ (top) and $\epsilon=\frac{1}{2}T$ (bottom).
  • ...and 2 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['t2']}
  • Theorem 2.5
  • ...and 13 more