Direct Algorithms for Reconstructing Small Conductivity Inclusions in Subdiffusion

TL;DR

This work tackles locating small conductivity inclusions within a homogeneous medium under a Caputo-based subdiffusion model by boundary measurements. It develops two direct, algebraic imaging algorithms grounded in the leading ε-expansion of boundary data and approximate Green functions, supported by a rigorous analysis of the expansion and stability when using truncated solutions. The methods yield accurate 2D/3D localization for circular and elliptical inclusions and demonstrate robustness to noise and limited data through harmonic- and Green-function-based reconstructions and a MUSIC-like indicator for multiple inclusions. The results offer computationally cheap, non-iterative alternatives to regularized inverse methods for subdiffusion conductivity imaging with theoretical guarantees and practical efficacy.

Abstract

The subdiffusion model that involves a Caputo fractional derivative in time is widely used to describe anomalously slow diffusion processes. In this work we aim at recovering the locations of small conductivity inclusions in the model from boundary measurement, and develop novel direct algorithms based on the asymptotic expansion of the boundary measurement with respect to the size of the inclusions and approximate fundamental solutions. These algorithms involve only algebraic manipulations and are computationally cheap. To the best of our knowledge, they are first direct algorithms for the inverse conductivity problem in the context of the subdiffusion model. Moreover, we provide relevant theoretical underpinnings for the algorithms. Also we present numerical results to illustrate their performance under various scenarios, e.g., the size of inclusions, noise level of the data, and the number of inclusions, showing that the algorithms are efficient and robust.

Figures (10)

• Figure 1: Illustration of the algorithm for locating one circular inclusion. The black solid dot denotes the target inclusion. The solutions $P_1$ and $P_2$ to \ref{['eq:reconst:bytwolines']} are located at the blue $+$ marks, and the reconstruction $P$ is indicated by the blue solid dot. The red line segments in (a) and the interiors of red circles in (b) indicate the orthogonal projections of $\Omega$ onto $\Sigma_1$ and $\Sigma_2$.
• Figure 1: \newlabelfig:noisy:reconstruction0 The reconstruction results for Example \ref{['ex:1']} with four configurations of one inclusion $(z_1,\varepsilon)$. Any pair of two non-parallel red line segments serves as the pair of domains $(\widetilde{\Sigma}_1,\widetilde{\Sigma}_2)$. The markers denote the solution $P_j$ to \ref{['eq:reconst:bytwolines']}.
• Figure 2: \newlabelfig:noisy:rootfinding0 The impact of different noise levels on measurement data for Example \ref{['ex:1']}. The top and bottom panels show the results for $(z_1,\varepsilon)=((0.2,0.3),0.05)$ and $(z_1,\varepsilon)=((0.2,0.3),0.1)$, respectively. Panels (a) and (b) show $\{u, U\}$ and $\widetilde{u} - U$ on $\partial\Omega$ at $T = 1$ with $U=U_1$, respectively, where $\partial\Omega$ is parameterized by $(\cos s,\sin s)$. Panels (c) and (d) show the value of $I_{\Phi}(U)$ for $U = U_1$ and $U = U_2$, respectively, with different noise levels, where for $U = U_j$, $\Phi = \Psi_{(P_j,0),3}(x,T-t)$, $j = 1, 2$, with $P_1 = (x_1, 2)$ and $P_2 = (2, x_2)$.
• Figure 3: \newlabelfig:one:ellipse0 The reconstruction results for Example \ref{['ex:1:variant']} with three aspect ratios $\varrho=2$, $3$ and $5$. Any pair of two non-parallel red line segments serves as the pair of domains $(\widetilde{\Sigma}_1, \widetilde{\Sigma}_2)$. The markers denote the solution $P_j$ to \ref{['eq:reconst:bytwolines']}.
• Figure 4: \newlabelfig:one:ellipse:u0 $u - U$ for Example \ref{['ex:1:variant']} on $\partial\Omega$ at $T=1$, with $\partial\Omega$ parametrized by $(\cos s,\sin s)$.

Theorems & Definitions (31)

• Theorem 1
• Lemma 2
• Proof 1
• Lemma 3
• Proof 2
• Lemma 4
• Proof 3
• Proof 4: Proof of Theorem \ref{['thm:asymptotic-exp']}
• Lemma 1
• Lemma 2