The approximate inverse for solving an inverse scattering problem for acoustic waves in an inhomogeneous medium

An application of the method of approximate inverse to a two-dimensional inverse scattering problem is discussed. The determination of the refractive index from an inverse scattering experiment is a nonlinear inverse problem. In this paper we treat the 2D scalar case. We propose to split this nonlinear problem into an ill-posed linear problem and a (well-posed) nonlinear part by first solving the data equation for the induced sources, consisting of the product of the refractive index and the field inside the object. This procedure retains the nonlinear relation between the two unknowns and treats it implicitly. The linear problem is efficiently solved by applying the method of approximate inverse. A reconstruction kernel is precomputed via the singular-value decomposition of the scattering operator. The nonlinear part is solved by treating the object equation. The use of the method of approximate inverse makes it possible to determine the refractive index and to locate inhomogeneities in the inverse medium problem. Numerical results are presented for a number of representative objects.