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A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data

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    * Corresponding author 
This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-15-1-0233 and by the Office of Naval Research grant N00014-15-1-2330
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  • The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any a priori knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

    Mathematics Subject Classification: 35R30, 78A46, 65C20.


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  • Figure 1.  For $k = 6.48$, we present in (a) the absolute value of the noisy backscattered field on the rectangle $(-5, 5)^2\times \{z = -7.6\}$ and in (b) the absolute value of the propagated data on the rectangle $(-5, 5)^2\times \{z = -0.75 \}$

    Figure 2.  Visualizations of the exact coefficient $c(\mathbf{x})$ in (120) (left) and the reconstructed coefficient $c_{comp}(\mathbf{x})$ (right) for the case of complete data with 15% artificial noise. The first row is the projection of $c(\mathbf{x})$ and $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The last row is a 3D isosurface, with isovalue 2.45, of the exact and reconstructed geometry of the target using MATLAB

    Figure 3.  Reconstruction result for the coefficient $c(\mathbf{x})$ in (120) with backscatter data. The left picture is the projection of $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The right one is the reconstructed geometry of the target

    Figure 4.  Visualizations of the exact coefficient $c(\mathbf{x})$ (left) in (123) and the reconstructed coefficient $c_{comp}(\mathbf{x })$ (right) for the case of backscatter data. The first row is the projection of $c(\mathbf{x})$ and $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The last row is a 3D visualization of the exact and reconstructed geometry of the target using MATLAB's isosurface. The isovalue is chosen as 50% of the maximal value of $c_{comp}(\mathbf{x})$

    Figure 5.  Visualizations of exact (left) and reconstructed (right) geometry of the target using the isosurface command in MATLAB

    Table 1.  Measured and computed dielectric constants $c$ of the targets

    Target Measured $c$ (std. dev.) Computed $c_{\max }$ Relative error
    A piece of yellow pine 5.30 (1.6%) 5.44 2.6%
    A piece of wet wood 8.48 (4.9%) 7.60 10.3%
    A geode 5.44 (1.1%) 5.55 2.0%
    A tennis ball 3.80 (13.0%) 4.00 5.2%
    A baseball not available 4.76 n/a
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