Inverse Fluid-solid Interaction Scattering Problem Using Phased and Phaseless Far Field Data

Abstract

We consider the inverse fluid-solid interaction scattering of incident plane wave from the knowledge of the phased and phaseless far field patterns. For the phased data, one direct sampling method for location and shape reconstruction is proposed. Only inner product is involved in the computation, which makes it very simple and fast to be implemented. With the help of the factorization of the far field operator, we give a lower bound of the proposed indicator functional for the sampling points inside the elastic body. While for the sampling points outside, we show that the indicator functional decays like the Bessel function as the points go away from the boundaries of the elastic body. We also show that the proposed indicator functional continuously dependents on the far field patterns, which further implies that the novel sampling method is extremely stable with respect to data error. For the phaseless data, to overcome the translation invariance, we consider the scattering of point sources simultaneously. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Numerically, we introduce a phase retrieval algorithm to retrieve the phased data without the additional obstacle. The novel phase retrieval algorithm can also be combined with the sampling method for phased data. We also design two novel direct sampling methods using the phaseless data directly. Finally, some numerical simulations in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed numerical methods.