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In this thesis we consider an impenetrable cylindrical scatterer surrounded by a heterogeneous dielectric coating. We present a numerical scheme for reconstructing the refractive index of the heterogeneous medium using far field data. The numerical scheme is based only on the mild assumption that the inhomogeneous medium is contained inside a circular cylinder, and does not require axis-symmetry or other similar restrictions. Moreover, we only require the boundary of the scatterer to be piecewise smooth. We reformulate this inverse problem as a nonlinear equation, which we then solve using a regularised Newton-type solver. The key innovation is performing nonlinear function evaluations, which involve solving a forward scattering problem, using an effcient coupled finite-element/boundary element method (FEMBEM) for the heterogeneous Helmholtz equation, which ensures that the important radiation condition is incorporated exactly. We derive an analytic representation for the Fréchet derivative for the heterogeneous dielectric coated scatterer, which we effciently compute using a novel extension of the coupled FEM-BEM scheme. The scheme is then demonstrated by reconstructing challenging continuous and discontinous media from noisy far field data.