Reconstruction of shapes and impedance functions using few far-field measurements

We consider the reconstruction of complex obstacles from few farfield acoustic measurements. The complex obstacle is characterized by its shape and an impedance function distributed along its boundary through Robin type boundary conditions. This is done by minimizing an objective functional, which is the L distance between the given far field information g∞ and the far field of the scattered wave u∞ corresponding to the computed shape and impedance function. We design an algorithm to update the shape and the impedance function alternatively along the descent direction of the objective functional. The derivative with respect to the shape or the impedance function involves solving the original Helmholtz problem and the corresponding adjoint problem, where boundary integral methods are used. Further we implement level set methods to update the shape of the obstacle. To combine level set methods and boundary integral methods we perform a parametrization step for a newly updated level set function. In addition since the computed shape derivative is defined only on the boundary of the obstacle, we extend the shape derivative to the whole domain by a linear transport equation. Finally, we demonstrate by numerical experiments that our algorithm reconstruct both shapes and impedance functions quite accurately for non-convex shape obstacles and constant or non-constant impedance functions. The algorithm is also shown to be robust to the initial guess of the shape, the initial guess of the impedance function and even large percentage noise. keywords Inverse Scattering, Far Field Pattern Data, Robin Type Boundary Conditions, Boundary Integral Methods, Level Set Methods, Shape Derivative.