Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement

Figure 1.  A radial problem in 2D with $ \mu_{1}^{\ast} = 1.2 $ and source function $ f = 1 $

Figure 2.  2D radial problem with source function $ f = 1 $ and varying $ \mu_{1}^{\ast} $

Figure 3.  Results for a 2D radial problem with $ f(x) = \delta(x) $

Figure 4.  Results for a 2D radial problem with $ f(x) = \delta(x) $ and varying $ \eta_{a}^{\ast} $

Figure 5.  Results for a 2D radial problem with $ f(x) = \delta(x) $ under noisy data, with $ \eta^{\ast} = 0.2 $ (left three plots) and $ \eta^{\ast} = 5 $ (right three plots)

Figure 6.  An example featuring a non-circular boundary interface

Figure 7.  An example featuring a non-circular boundary interface with noisy data

Figure 8.  Results for a non-circular boundary interface with varying $ \eta_{a}^{\ast} $ values

Figure 9.  An example featuring a non-circular boundary interface with noisy data

Figure 10.  An example of a peanut-shaped boundary interface with noisy data

Figure 11.  Effect of choice of initial guess

Figure 12.  Reconstruction results for a non-circular interface with $ f(x) = 100\delta(x) $ using exact measurements (top row) and noisy measurements ($ \gamma = 0.01 $, bottom row)

Figure 13.  Results for a smooth non-circular boundary interface with $ f(x) = 100\delta(x) $ and $ \mu_{1}^{\ast} = 6 $ under exact (first three columns from the left) and noisy measurements ($ \gamma = 0.01 $)

Figure 14.  Results for a smooth non-circular boundary interface with $ f(x) = 100\delta(x) $ and $ \mu_{1}^{\ast} = 6 $ under exact and noisy measurements ($ \gamma = 0.005 $)

Figure 15.  Results for a square boundary interface

Figure 16.  Results for an inverted $ \textsf{T}$-shaped boundary interface

Figure 17.  Results for a circular boundary interface with $ f $ (given by (38) with $ \epsilon = 0.5 $) of various positions and instances near the boundary and parameters $ (\mu_{0}^{\ast}, \mu_{1}^{\ast}) = (1, 1.2) $, under exact (first three columns from the left) and noisy measurements. No perimeter regularization was applied in any of the cases

Figure 18.  Results for a peanut-shape boundary interface with source $ f $ (given by (38) with $ \epsilon = 0.5 $) near the boundary, under exact (top row) and noisy measurements. Perimeter penalization was applied in all of the cases with $ \rho_{1} = 0.00004 $

Figure 19.  Results for a peanut-shape boundary interface with source $ f $ (given by (38) with $ \epsilon = 0.3 $) near the boundary, under exact (top row) and noisy measurements. Perimeter penalization was applied in all of the cases with $ \rho_{1} = 0.00003 $

Figure 20.  Results for a peanut-shape boundary interface with source $ f $ (given by (38) with $ \epsilon = 0.2 $) near the boundary, under exact (top row) and noisy measurements. The locations of the point sources are marked by thick black dots. Perimeter penalization was applied in all of the cases with $ \rho_{1} = 0.00003 $

Figure 21.  Positioning of sources

Figure 22.  Results for a peanut-shape boundary interface with $ f $ (given by (38) with $ \epsilon = 0.5 $) of various positions and under noisy measurements. Perimeter penalization was applied in all of the cases with $ \rho_{1} = 0.000006 $