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Numerical reconstructions of a source term in a mobile-immobile diffusion model from the partial interior observation

Zhiwei Yang, Yikan Liu

TL;DR

This work addresses inverse source identification in a two-time-scale mobile-immobile time-fractional diffusion model governed by $(\partial_t u + q\,\partial_t^\alpha u + \mathcal{A} u)(\mathbf{x},t) = \rho(t) g(\mathbf{x})$ with Dirichlet boundaries. It develops a theoretical foundation using fractional Duhamel's principle and a weak vanishing property to establish uniqueness from partial interior data, and then formulates a regularized optimal control problem to recover the spatial source $g(\mathbf{x})$ from interior observations. A coupled forward–adjoint system is derived, yielding the gradient $\mathcal{J}'(g)=\int_0^T \rho(t) v(\mathbf{x},t)\,dt + \beta g(\mathbf{x})$, where $v$ solves an adjoint PDE with a backward fractional operator. A finite element conjugate gradient algorithm is implemented, solving forward and adjoint problems within each iteration to update $g$. Numerical experiments demonstrate robust, accurate reconstructions under various noise levels, observation domains, and fractional orders, validating the method's effectiveness for practical multi-time-scale diffusion problems.

Abstract

We consider an inverse source problem in the two-time-scale mobile-immobile fractional diffusion model from partial interior observation. Theoretically, we combine the fractional Duhamel's principle with the weak vanishing property to establish the uniqueness of this inverse problem. Numerically, we adopt an optimal control approach for determining the source term. A coupled forward-backward system of equations is derived using the first-order optimality condition. Finally, we construct a finite element conjugate gradient algorithm for the numerical inversion of the source term. Several experiments are presented to show the utility of the method.

Numerical reconstructions of a source term in a mobile-immobile diffusion model from the partial interior observation

TL;DR

This work addresses inverse source identification in a two-time-scale mobile-immobile time-fractional diffusion model governed by with Dirichlet boundaries. It develops a theoretical foundation using fractional Duhamel's principle and a weak vanishing property to establish uniqueness from partial interior data, and then formulates a regularized optimal control problem to recover the spatial source from interior observations. A coupled forward–adjoint system is derived, yielding the gradient , where solves an adjoint PDE with a backward fractional operator. A finite element conjugate gradient algorithm is implemented, solving forward and adjoint problems within each iteration to update . Numerical experiments demonstrate robust, accurate reconstructions under various noise levels, observation domains, and fractional orders, validating the method's effectiveness for practical multi-time-scale diffusion problems.

Abstract

We consider an inverse source problem in the two-time-scale mobile-immobile fractional diffusion model from partial interior observation. Theoretically, we combine the fractional Duhamel's principle with the weak vanishing property to establish the uniqueness of this inverse problem. Numerically, we adopt an optimal control approach for determining the source term. A coupled forward-backward system of equations is derived using the first-order optimality condition. Finally, we construct a finite element conjugate gradient algorithm for the numerical inversion of the source term. Several experiments are presented to show the utility of the method.
Paper Structure (8 sections, 5 theorems, 46 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 8 sections, 5 theorems, 46 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary and the theoretical uniqueness
  3. Numerical scheme for solving the direct problem (\ref{['main:e0']})
  4. Inverse problem and its conjugate gradient algorithm
  5. Mathematical formulation of the inverse problem and the corresponding inverse algorithm
  6. The reconstruction algorithm: finite element conjugate gradient algorithm
  7. Numerical experiments
  8. Concluding remarks

Key Result

Lemma 1

(i) Under assumption eq-assume, there exists a unique solution to the initial-boundary value problem main:e0. (ii) Let $g=0$ in $\Omega$ and $a\in L^2(\Omega)$. Then there exists a unique solution to main:e0. Moreover, the solution $u:(0,T)\longrightarrow H^2(\Omega)\cap H_0^1(\Omega)$ can be analytically extended to $(0,\infty)$.

Figures (3)

  • Figure 1: Illustration of the domain.
  • Figure 2: We fix the order $\alpha=0.5$ and present the numerical reconstruction for the inverse source of the membrane. The observable subdomain $\omega=\Omega\setminus[0.05,0.95]^2$, $T=1.5$ and the noise level $\epsilon=1$ in \ref{['noise_free']}. The first row from left to right are the reconstructed and the true solutions, respectively. The second row represents the absolute error and the corresponding loss with respect to the iterations.
  • Figure 3: We fix the order $\alpha=0.5$ and present the numerical reconstruction for the inverse source of the membrane. The observable subdomain $\omega=\Omega\setminus[0.05,0.95]^2$, $T=1.5$ and the noise level $\epsilon=1$ in \ref{['noise_free']}. The first row to the third row from left to right are the reconstructed and true solutions, respectively.

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • Example 2