Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization
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We consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at receivers outside the object. The data is assumed to be generated by plane waves impinging on the obstacle from multiple directions and at multiple frequencies. This inverse problem is reformulated as the optimization problem of finding band-limited shape and impedance functions which minimize the L^2 distance between the computed value of the scattered field at the receivers and the data. The optimization problem is non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate. The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of a domain or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions. The RLA is a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton's method at the new frequency. We demonstrate the effectiveness of the method with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are from sound-hard or sound-soft objects. While the method is effective in obtaining high quality reconstructions for complicated geometries and impedance functions, a number of interesting open questions remain. We present numerical experiments that suggest underlying mechanisms of success and failure, showing areas where improvements could help lead to robust and automatic tools.
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