Determining a Time-Varying Potential in Time-Fractional Diffusion from Observation at a Single Point

TL;DR

The paper addresses recovering a time-varying potential $\rho(t)$ in a time-fractional diffusion model from a single-point observation. By leveraging the smoothing properties of the direct problem and a weighted $L^p$ framework, it proves conditional Lipschitz stability and constructs a practical fixed-point reconstruction algorithm with a rigorous error analysis for both the forward discretization and the inverse reconstruction. The fully discrete scheme combines backward-Euler convolution quadrature with finite element spatial discretization, yielding provable error bounds and a contraction mapping for the inverse problem in weighted $\ell^p$ spaces. Numerical experiments in 1D corroborate the theoretical results, show robust convergence with noisy data, and illustrate the impact of the fractional order $\alpha$ and discretization parameters on reconstruction quality.

Abstract

We discuss the identification of a time-dependent potential in a time-fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by using the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.

Figures (6)

• Figure 1: The spatial convergence for $\alpha =0.25, 0.5$, and $0.75$, with exact observational data. The black dashed line is the plot for $O(h^2)$ convergence rate.
• Figure 2: The temporal convergence for $\alpha =0.25, 0.5$, and $0.75$, with exact observational data. The black dashed line is the plot for $O(\tau^{0.5})$ convergence rate.
• Figure 3: The convergence with respect to noise level $\delta$ for $\alpha =0.25, 0.5$ and $0.75$. The approximation is obtained by setting the discretization parameters $h$ and $\tau$ according to \ref{['eqn:optimal']}. The black dashed line is the plot for $O(\delta^{1/2})$ convergence rate.
• Figure 4: The reconstruction results from the noisy and noisy-free data, with $\alpha=0.5$. Taking optimal time step size $\tau=T/2^8$ when $T=0.5$. The lines in red indicate the exact potentials, lines in blue are reconstructions from the noisy-free data, and the green lines are reconstructions from the noisy data.
• Figure 5: The decay of error throughout the iterations. $k$ denotes the number of iterations. The first row: errors in $\ell^p$. The second row: errors in $\ell^p_\omega$ with $\omega=10$. The first column: errors from the exact data. The second column: errors from the noisy data.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 3.2
• proof
• Proposition 3.1
• proof
• Remark 3.1
• Lemma 4.1
• Lemma 4.2