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We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number M and the diameter a of the obstacles as well as the minimum distance d between them under which the Foldy-Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as is done sometimes in the literature, then this condition is (M −1) a d 2 < c, with an appropriate constant c, while if we use double layer potentials, then a weaker condition of the form √ M − 1 a d < c is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters M, a, and d. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering. B 1 , B 2 , . . . , B M be M open, bounded, and simply connected sets in R 3 with Lipschitz boundaries 1 containing the origin. We assume that the Lipschitz constants of B j , j = 1, . . . , M, are uniformly bounded. We set D m := B m + z m to be the small bodies characterized by the parameter > 0 and the locations z m ∈ R 3 , m = 1, . . . , M. Let U i be a solution of the Helmholtz equation (Δ + κ 2 )U i = 0 in R 3 . We denote by U s the acoustic field scattered by the M small bodies D m ⊂ R 3 due to the incident field U i . We restrict ourselves to (1) the plane incident waves, U i (x, θ) := e ikx·θ , with the incident direction θ ∈ S 2 , with S 2 being the unit sphere, and (2) the scattering by rigid bodies. Hence the total field U t := U i + U s satisfies the following exterior Dirichlet problem of the acoustic waves: Introduction and statement of the results. Let

In this paper, we investigate the problem of reconstructing soundsoft acoustic obstacles using multifrequency far field measurements corresponding to one direction of incidence. The idea is to obtain a rough estimate of the obstacle's shape at the lowest frequency using the least-squares approach, then refine it using a recursive linearization algorithm at higher frequencies. Using this approach, we show that an accurate reconstruction can be obtained without requiring a good initial guess. The analysis is divided into three steps. Firstly, we give a quantitative estimate of the domain in which the least-squares objective functional, at the lowest frequency, has only one extreme (minimum) point. This result enables us to obtain a rough approximation of the obstacle at the lowest frequency from initial guesses in this domain using convergent gradient-based iterative procedures. Secondly, we describe the recursive linearization algorithm and analyze its convergence for noisy data. We qualitatively explain the relationship between the noise level and the resolution limit of the reconstruction. Thirdly, we justify a conditional asymptotic Hölder stability estimate of the illuminated part of the obstacle at high frequencies. The performance of the algorithm is illustrated with numerical examples.

We deal with the problem of reconstruction of the coefficient discontinuities (or supports) of scalar divergence form equations with lower order terms from the Dirichlet-to-Neumann map using complex geometrical optics (CGO) solutions. We consider both penetrable and impenetrable obstacles. The usual proofs for justifying this method assume, in addition to the smoothness of the coefficients and the interfaces, the following two conditions. The finiteness of the touching points of the phase’s level curves (or surfaces) of the used CGO solutions with the interface. The positivity of the lower bound of the relative curvature of the interface. In this paper, we show how we can remove these two conditions and justify the reconstruction method considering L∞ coefficients and Lipschitz interfaces of discontinuity.