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Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

Siyu Cen, Bangti Jin, Yavar Kian, Eric Soccorsi, Rachid Zarouf, Zhi Zhou

Abstract

In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.

Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

Abstract

In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.
Paper Structure (12 sections, 6 theorems, 64 equations, 7 figures, 3 tables)

This paper contains 12 sections, 6 theorems, 64 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main results and discussions
  3. Analysis of the direct problem
  4. Preliminaries
  5. Well-posedness
  6. Time-analyticity
  7. Proof of Theorems \ref{['t1']} and \ref{['t2']}
  8. Proof of Theorem \ref{['t1']}
  9. Proof of Theorem \ref{['t2']}
  10. Numerical results and discussions
  11. Levenberg-Marquardt method
  12. Numerical results and discussions

Key Result

Theorem 2.1

For $j=1,2$, let $\alpha_j \in(0,1)$, $\ell_j \in (0,\infty)$, $q_j\in L^\infty(0,\ell_j)$ be non-negative, and $\rho_j\in L^\infty(0,\ell_j)$ be piecewise constant and fulfill condition rho with $\rho=\rho_j$. Let $g\in W^{1,1}(0,T)$ be not everywhere zero and satisfy $g(0)=0$ and Denote by $u_j$, $j=1,2$, the solution to eq1 with $(\alpha,\ell,q,\rho)=(\alpha_j, \ell_j,q_j, \rho_j)$. Then the m

Figures (7)

  • Figure 1: The convergence of the algorithm for Example \ref{['ex:Dirichlet']}, at three noise levels, $\alpha=0.75$.
  • Figure 2: The reconstructions of the potential $q$ for Example \ref{['ex:Dirichlet']}.
  • Figure 3: The singular values of the Jacobian $\partial_q F$ in Examples \ref{['ex:Dirichlet']}, \ref{['ex:Neumann']}, and \ref{['ex:Dirichlet_unknown_rho']} (from left to right).
  • Figure 4: The convergence of the algorithm for Example \ref{['ex:Neumann']} at three noise levels, for $\alpha=0.75$.
  • Figure 5: The reconstructions of the potential $q$ for Example \ref{['ex:Neumann']} with $1\%$ noise.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 5 more