Factorization method for the biharmonic scattering problem for an absorbing penetrable scatterer

Figure 1.  Exact and Born-approximated far-field matrices for scattering by a disk of radius $ \epsilon = 0.5 $ with complex refractive index $ n = 1.0+0.5\text{i} $ at wavenumber $ \kappa = 2 $. The comparison illustrates the close approximation with a slight deviation of the Born approximation from the exact solution in the presence of moderate absorption

Figure 2.  Comparison of the reconstructed penetrable disk $ D = B_{\epsilon} $ using far-field data for the disk scatterer with radius $ \epsilon = 0.5 $, wavenumber $ \kappa = 2 $, and $ n = 1.0+0.5\text{i} $

Figure 3.  Absolute error $ \|\mathbf F_{\rm exact} - \mathbf F_{\rm Born}\|_\infty $ between the exact and Born far-field data. Left: Error as a function of the wavenumber $ \kappa $ for a single disk of fixed radius $ \epsilon = 0.5 $ and fixed index of refraction $ n = 1.0+1.5\text{i} $. Right: Error as a function of the contrast $ |n-1| $ for a fixed wavenumber $ \kappa = 2 $ and a fixed radius $ \epsilon = 0.5 $

Figure 4.  Reconstruction of the star–shaped penetrable scatterer using the factorization method for two different wavenumbers: $ \kappa = 2\pi $ (left) and $ \kappa = 3\pi $ (right) with a constant contrast $ n = 2.5+0.5\text{i} $

Figure 5.  Reconstruction of the star–shaped penetrable scatterer using the factorization method for two imaginary refractive indices with a fixed wavenumber $ \kappa = \pi $: Left: $ n = 0.1\text{i} $ and Right: $ n = 0.3\text{i} $

Figure 6.  Reconstruction of the star–shaped penetrable scatterer using the factorization method for $ \kappa = 2\pi $ with $ n = 2.5+0.5\text{i} $ and noise added. Left: 5% noise ($ \delta = 0.05 $). Right: 10% noise ($ \delta = 0.10 $)

Figure 7.  Reconstruction of the shifted star–shaped penetrable scatterer using the factorization method for $ \kappa = 2\pi $ with $ 5\% $ noise and index of refraction $ n = 2.5+0.5\text{i} $. Left: Scatterer shifted to $ (-0.5, 0.5) $. Right: Scatterer shifted to $ (1, 1.5) $

Figure 8.  Reconstruction of the penetrable kite–shaped scatterer using the factorization method for two different wavenumbers: $ \kappa = \pi $ (left) and $ \kappa = 4\pi $ (right) with a contrast of $ n = 2.0+0.1\text{i} $

Figure 9.  Reconstruction of the penetrable kite–shaped scatterer using the factorization method for two imaginary refractive indices with a fixed wavenumber $ \kappa = \pi $: Left: $ n = 1.0\text{i} $ and Right: $ n = 3.0\text{i} $

Figure 10.  Reconstruction of the penetrable kite–shaped scatterer using the factorization method for $ \kappa = \pi $ with refractive index $ n = 2.0+0.1\text{i} $ and noise added. Left: 10% noise ($ \delta = 0.10 $). Right: 20% noise ($ \delta = 0.20 $)

Figure 11.  Reconstruction of the shifted kite–shaped penetrable scatterer using the factorization method for $ \kappa = \pi $ with $ 10\% $ noise and index of refraction $ n = 2.0+0.1\text{i} $. Left: Scatterer shifted to $ (-1, 0) $. Right: Scatterer shifted to $ (0, 1) $

Figure 12.  Reconstruction of two penetrable disks with distinct radii $ \epsilon_1 = 0.25 $ and $ \epsilon_2 = 0.50 $ and refractive indices $ n_1 = 1 + 2.0\mathrm{i} $ and $ n_2 = 2 + 1.0\mathrm{i} $ using the factorization method with $ \kappa = \pi $. Left: No noise. Right: $ 5\% $ noise ($ \delta = 0.05 $)