A coupled FEM-BEM algorithm for the inverse acoustic medium problem

We present a numerical scheme for reconstructing the refractive index of an inhomogeneous two dimensional medium using acoustic far field data. The numerical scheme is based only on the mild assumption that the inhomogeneous medium is contained in the unit disk, and does not require axis-symmetry or other similar restrictions. Reconstruction of the refractive index, without the assumption of axis-symmetry, is achieved using an expansion in the high order Logan-Shepp polynomials. The Logan-Shepp expansion coefficients of the refractive index are formulated as the solution of a nonlinear equation, which is solved using a regularised Newton-type solver. Nonlinear function evaluations, which involve solving a forward scattering problem, are performed using an efficient coupled finite-element/boundary element method, which ensures that the radiation condition is incorporated exactly. The scheme is demonstrated by reconstructing challenging continuous and discontinous media from noisy far field data.