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

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\% $