On the Kohn–Vogelius formulation for solving an inverse source problem

Figures & data

Figure 1. Example 1: Three balls, on-grid, no noise. (A) Target. (B) Result.

Figure 2. Example 2: Rectangular target, no noise, on grid.

Figure 3. Example 2: Reconstruction of one rectangular target, no noise, on grid. (A) One ball. (B) Two balls.

Table 1. Example 2: Error in constructing one rectangular target from one ball, no noise, on grid.

Distance function	$L^2({\mathrm{\Gamma }}_M)$-norm	$L^2(\mathrm{\Omega })$-norm	$H^1(\mathrm{\Omega })$-seminorm	$H^1(\mathrm{\Omega })$-norm
% Error in equivalent radius	9.0136	0.8125	9.0398	9.0361
Distance from rectangle's centroid to ball's centre	$6.94\times {10}^{-2}$	$1.03\times {10}^{-5}$	$6.94\times {10}^{-2}$	$6.94\times {10}^{-2}$

Table 2. Example 2: Error in constructing one rectangular target from two balls, no noise, on grid.

Distance function	$L^2({\mathrm{\Gamma }}_M)$-norm	$L^2(\mathrm{\Omega })$-norm	$H^1(\mathrm{\Omega })$-seminorm	$H^1(\mathrm{\Omega })$-norm
% Error in equivalent radius	0.1787	0.1783	0.1798	0.1797
Distance from rectangles' compound centroid to ball's centre	$4.13\times {10}^{-3}$	$4.11\times {10}^{-3}$	$4.16\times {10}^{-3}$	$4.16\times {10}^{-3}$

Figure 4. Example 3: Target for one ball, no noise.

Figure 5. Example 3: Result for one ball, no noise, with varying grid densities.

Figure 6. Example 4: Target for one ball corrupted with varying levels of noise μ. (A) $\mu =5\mathrm{\%}$. (B) $\mu =10\mathrm{\%}$. (C) $\mu =15\mathrm{\%}$. (D) $\mu =20\mathrm{\%}$. (E) $\mu =25\mathrm{\%}$. (F) $\mu =30\mathrm{\%}$. (G) $\mu =35\mathrm{\%}$. (H) $\mu =40\mathrm{\%}$. (I) $\mu =45\mathrm{\%}$. (J) $\mu =50\mathrm{\%}$.

Figure 7. Example 4: Result for one ball, on-grid, with varying levels of noise.

Figure 8. Example 5: Result for one ball, off-grid, with varying levels of noise.