Article Contents
Article Contents

# On the reconstruction of medium conductivity by integral equation method based on the Levi function

• *Corresponding author: Jijun Liu

This work is supported by National Key R&D Program of China (No.2020YFA0713800), NSFC(No.11971104). The second author thanks Prof. G.Nakamura in Hokkaido university for heuristic discussions

• Consider an inverse problem of recovering the medium conductivity governed by an elliptic system, with partial information of the solution specified in some internal domain as inversion input. We firstly establish the uniqueness of this inverse problem and the conditional stability of H$\ddot{\text{o}}$lder type in internal domain in terms of the analytic extension of the solution. Then by representing the solution of the direct problem with variable coefficient under the Levi function framework, this nonlinear inverse problem is reformulated as solving a linear integral system provided that the boundary value of the conductivity be known. Then this linear system is regularized to deal with the ill-posedness of the function extension, with an efficient numerical realization scheme for seeking the regularizing solution firstly for the density pair and then for the conductivity to be recovered. Numerical implementations are presented to show the validity of the proposed scheme.

Mathematics Subject Classification: Primary: 35R25, 35R30, 65N21, 65N30, 65R20; Secondary: 35B30.

 Citation:

• Figure 1.  The exact solution of $\sigma_e$(left) and $u_e$(right).

Figure 2.  Geometric setting

Figure 3.  Extension of $u$ from $\Omega_0$ to $\Omega$: exact solution (left), extended one (middle), absolute error distribution with average relative error $\text{RE}^{\Omega}(u^e, u^n) = 0.0065$ (right)

Figure 4.  Contrast of exact $-u_x$(left) and $B_x$(middle), and the absolute error(right), with $r_0 = 0.8$, $r_{\epsilon_0} = 0.9$

Figure 5.  Contrast of exact $-u_y$(left) and $B_y$(middle), and the absolute error(right), with $r_0 = 0.8$, $r_{\epsilon_0} = 0.9$

Figure 6.  Contrast of exact $-\Delta u$(left) and density function $\tilde{\mu}$(middle), and the absolute error(right), with $r_0 = 0.8$, $r_{\epsilon_0} = 0.9$

Figure 7.  Recovery of $\sigma$ in $\Omega_{\epsilon_0}$ using exact interior data in $\Omega_0$: exact conductivity(left), recovered one(middle), absolute error with average relative error $\text{RE}^{\Omega_{\epsilon_0}}(\sigma^e, \sigma^n) = 0.2034$(right)

Figure 8.  Relative error of $(u,\sigma)$ with different noise levels $\delta$ under fixed $\beta = 5e-6$

Figure 9.  Recovery of $\sigma$ with different noise levels for $\beta = 5e-6$: (A) Exact distribution. (B) Recovered one for $\delta = 0$. (C) Recovered one for $\delta = 2\%$. (D) Recovered one for $\delta = 4\%$

Figure 10.  Numerical solutions $B_x$(first row), $B_y$(second row) and $\tilde{\mu}$(third row) with different $r_0$ for fixed $r_{\epsilon_0} = 0.85$: Column (I) $r_0 = 0.8$. (J) $r_0 = 0.70$. (K) $r_0 = 0.60$. (L) $r_0 = 0.50$

Figure 11.  Numerical solutions $\sigma_n$ for different $r_0$ with fixed $r_{\epsilon_0} = 0.85$: (A) $r_0 = 0.8$. (B) $r_0 = 0.70$. (C) $r_0 = 0.60$. (D) $r_0 = 0.50$.

Figure 12.  The exact solution of conductivity $\sigma_e$(left) and voltage $u_e$(right)

Figure 13.  Recovery of $u$ from exact inversion input given in $\Omega_0$: exact value(left), recovered one(middle), and absolute error distribution(right) with $\text{RE}^{\Omega}(u^e, u^n) = 0.0010$

Figure 14.  Contrast of exact $-u_x$(left) and $B_x$(middle), the absolute error(right)

Figure 15.  Contrast of exact $-u_y$(left) and $B_y$(middle), the absolute error(right)

Figure 16.  Contrast of exact $-\Delta u$(left) and $\tilde{\mu}$(middle), the absolute error(right)

Figure 17.  Recovery of $\sigma$ in $\Omega_{\epsilon_0}$ from exact inversion input data given in $\Omega_0$: exact $\sigma^e$(left), recovered one(middle), absolute error(right) with $\text{RE}^{\Omega_{\epsilon_0}}(\sigma^e, \sigma^n) = 0.1631$

Figure 18.  Relative error of $u$(left) and $\sigma$(right) with respect to $\delta$ for $\beta = 5e-6$

Figure 19.  Recovery of $\sigma$ with different noise levels for $\beta = 5e-6$: (A) Exact distribution. (B) Recovered one for $\delta = 0$. (C) Recovered one for $\delta = 4\%$. (D) Recovered one for $\delta = 6\%$

Table 1.  Error distribution of reconstructions in terms of noise levels

 $\delta$ 0 1% 2% 3% 4% 5% $\text{RE}^{\Omega}(u^e, u^{\beta, \delta})$ 0.0065 0.0098 0.0100 0.0102 0.0106 0.0107 $\text{RE}^{\Omega_{\epsilon_0}}(\sigma^e, \sigma^{\beta, \delta})$ 0.2034 0.2322 0.2488 0.2958 0.3195 1.0731

Table 2.  Relative errors for recovering $u$ for different sizes of $\Omega_0$

 $r_0$ 0.9 0.8 0.7 0.6 0.5 $\text{RE}^{\Omega}(u^e, u^n)$ 0.0019 0.0065 0.0158 0.0268 0.0416

Table 3.  Relative errors for recovering $\sigma$ for different configurations of $(\Omega_{\epsilon_0},\Omega_0)$

 0.75 0.8 0.85 0.9 0.8 0.1909 0.2244 0.2237 0.2034 0.7 0.3492 0.3279 0.3218 0.9721 0.6 0.4083 0.3652 0.4047 0.6686 0.5 0.405 0.3882 0.5429 0.9738

Table 4.  Relative errors of recovering $(u,\sigma)$ for different noise level $\delta$

 $\delta$ 0 1% 2% 3% 4% 5% $\text{RE}^{\Omega}(u^e, u^{\beta, \delta})$ 0.001 0.0022 0.0023 0.0023 0.0024 0.0027 $\text{RE}^{\Omega_{\epsilon_0}}(\sigma^e, \sigma^{\beta, \delta})$ 0.1631 0.2102 0.2111 0.2333 0.2777 0.3526

Table 5.  Relative errors for recovering $\sigma$ for different configurations of $(\Omega_0,\Omega_{\epsilon_0})$

 0.75 0.8 0.85 0.9 0.8 0.1894 0.2138 0.2004 0.1631 0.7 0.2665 0.2528 0.2167 0.2241 0.6 0.3098 0.2932 0.2616 0.3497 0.5 0.3483 0.3784 0.3186 0.4119
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