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Inverse source problem with a posteriori interior measurements for space-time fractional diffusion equations

Kai Yu, Zhiyuan Li, Yikan Liu

TL;DR

This work tackles the inverse source problem for a space-time fractional diffusion equation using a posteriori interior measurements. It establishes a uniqueness result by exploiting the memory effect of the fractional time derivative and a unique continuation property, with Duhamel's principle linking temporal activity to the spatial source. For reconstruction, the problem is recast as a Tikhonov-regularized optimization and solved by a Levenberg-Marquardt scheme, using a forward map that sends the spatial source $f$ to interior observations $u|_{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}}}}}}}$ and incorporating a discrepancy principle for stopping. Numerical experiments in 1D confirm accurate recovery of $f$ from noisy interior data and illustrate stability as the observation window is varied. The results demonstrate a practical methodology for solving a nonlocal inverse problem with potential applications to environmental monitoring and hazard assessment.

Abstract

This paper investigates an inverse source problem for space-time fractional diffusion equations from a posteriori interior measurements. The uniqueness result is established by the memory effect of fractional derivatives and the unique continuation property. For the numerical reconstruction, the inverse problem is reformulated as an optimization problem with the Tikhonov regularization. We use the Levenberg-Marquardt method to identity the unknown source from noisy measurements. Finally, we give some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm.

Inverse source problem with a posteriori interior measurements for space-time fractional diffusion equations

TL;DR

This work tackles the inverse source problem for a space-time fractional diffusion equation using a posteriori interior measurements. It establishes a uniqueness result by exploiting the memory effect of the fractional time derivative and a unique continuation property, with Duhamel's principle linking temporal activity to the spatial source. For reconstruction, the problem is recast as a Tikhonov-regularized optimization and solved by a Levenberg-Marquardt scheme, using a forward map that sends the spatial source to interior observations and incorporating a discrepancy principle for stopping. Numerical experiments in 1D confirm accurate recovery of from noisy interior data and illustrate stability as the observation window is varied. The results demonstrate a practical methodology for solving a nonlocal inverse problem with potential applications to environmental monitoring and hazard assessment.

Abstract

This paper investigates an inverse source problem for space-time fractional diffusion equations from a posteriori interior measurements. The uniqueness result is established by the memory effect of fractional derivatives and the unique continuation property. For the numerical reconstruction, the inverse problem is reformulated as an optimization problem with the Tikhonov regularization. We use the Levenberg-Marquardt method to identity the unknown source from noisy measurements. Finally, we give some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm.

Paper Structure

This paper contains 6 sections, 6 theorems, 50 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries
  3. Proof of Theorem \ref{['thm-isp']}
  4. Levenberg-Marquardt method
  5. Numerical experiments
  6. Conclusion

Key Result

Theorem 1.2

Let $\phi=0,$$f\in L^2(\Omega),$$\mu\in L^2(0,T)$ and $u$ be the solution to eq-gov. Assume $f=0$ in a nonempty open subdomain $\omega\subset\Omega$. If $\mu$ satisfies then for any $T\ge T_0$ and any $T_1\in(0,T),$$u=0$ in $\omega\times(T_1,T)$ implies $f=0$ in $\Omega$.

Figures (2)

  • Figure 1: Reconstruction of the source term $f_{\mathrm{true}}^1$ in $\Omega\setminus\omega$ with various noise levels.
  • Figure 2: Reconstruction of the source term $f_{\mathrm{true}}^2$ in $\Omega\setminus\omega$ with various noise levels.

Theorems & Definitions (9)

  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1: Unique continuation
  • proof
  • Lemma 3.2: Duhamel's principle
  • proof : Proof of Theorem \ref{['thm-isp']}