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A control-based spatial source reconstruction in fractional heat equations

Galina García, Joaquín Vidal, Sebastián Zamorano

TL;DR

This work tackles the inverse source problem for a nonlocal fractional heat equation by reconstructing the spatial factor $f$ in the separable source $F(x,t)=f(x)\sigma(t)$ from partial observations of the state and its time derivative on a subdomain. The authors derive a reconstruction formula for the Fourier coefficients of $f$ based on the null controllability of the fractional heat equation for $s\in(\tfrac{1}{2},1)$, spectral decompositions, and Volterra integral equations, enabling recovery with limited data. A penalized HUM optimal-control framework is developed to compute the necessary null controls and solve the associated Volterra problems, with a full numerical scheme provided. Numerical experiments demonstrate robust recovery of low- and moderate-frequency components, accurate support/shape capture, and controlled performance for discontinuous or high-frequency sources, highlighting the method’s potential for imaging and material sciences where nonlocal diffusion is relevant.

Abstract

This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state and its time derivative over a subset of the domain. A reconstruction formula for the Fourier coefficients of the unknown source is derived, leveraging the null controllability property of the fractional heat equation when the fractional order lies in the interval $s\in(1/2,1)$. The methodology builds on spectral analysis and Volterra integral equations, providing a robust framework for recovering spatial sources under limited measurement data. Numerical experiments confirm the accuracy and stability of the proposed approach.

A control-based spatial source reconstruction in fractional heat equations

TL;DR

This work tackles the inverse source problem for a nonlocal fractional heat equation by reconstructing the spatial factor in the separable source from partial observations of the state and its time derivative on a subdomain. The authors derive a reconstruction formula for the Fourier coefficients of based on the null controllability of the fractional heat equation for , spectral decompositions, and Volterra integral equations, enabling recovery with limited data. A penalized HUM optimal-control framework is developed to compute the necessary null controls and solve the associated Volterra problems, with a full numerical scheme provided. Numerical experiments demonstrate robust recovery of low- and moderate-frequency components, accurate support/shape capture, and controlled performance for discontinuous or high-frequency sources, highlighting the method’s potential for imaging and material sciences where nonlocal diffusion is relevant.

Abstract

This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state and its time derivative over a subset of the domain. A reconstruction formula for the Fourier coefficients of the unknown source is derived, leveraging the null controllability property of the fractional heat equation when the fractional order lies in the interval . The methodology builds on spectral analysis and Volterra integral equations, providing a robust framework for recovering spatial sources under limited measurement data. Numerical experiments confirm the accuracy and stability of the proposed approach.

Paper Structure

This paper contains 26 sections, 10 theorems, 138 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. The inverse problem
  3. State of the art
  4. Outline of the article
  5. Preliminary results
  6. Functional setting
  7. Fractional Laplacian
  8. Nonlocal heat equation
  9. Volterra integral equations
  10. Main results
  11. Optimal control problem: penalized HUM approach
  12. Numerical Reconstruction Scheme
  13. Numerical solution of the nonlocal heat equation
  14. Approximation of eigenfunctions
  15. Approximation with partial sums
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let $u\in H_0^s((-1,1))$ be such that $(-\partial_x^2)^s u\in L^2(-1,1)$. Then for every $v\in H^s({\mathbb{R}})$, the following identity holds holds.

Figures (17)

  • Figure 1: Eigenvalues estimated with FEM and approximation by \ref{['aproxeigenvalues']}.
  • Figure 2: Eigenvalues according to the size of the partition.
  • Figure 3: Eigenfunctions estimated with FEM and approximation by \ref{['aproxeigenfunctions']}.
  • Figure 4: Approximation of final state $\Phi_j(x,0)$ with FEM and Partial Sums.
  • Figure 5: Approximation of final state $\Phi_0(x,0)$ with FEM and exact formula.
  • ...and 12 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 5 more