Session S20 - Applied Math and Computational Methods and Analysis across the Americas

Friday, July 16, 13:00 ~ 13:30 UTC-3

Multifrequency inverse obstacle scattering with unknown impedance boundary conditions

Carlos Borges

Department of Mathematics at the University of Central Florida, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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. This inverse problem is reformulated as the optimization problem of finding band-limited shape and impedance functions which minimize the L2 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.

Joint work with Manas Rachh (Center for Computational Mathematics, Flatiron Institute, New York, USA).

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