(untitled) (Journal of Computational Mathematics, 2018)
This paper concerns the reconstruction of a penetrable obstacle embedded in a waveguide using the scattered data due to point sources, which is formulated as an optimization problem. We propose a fast reconstruction method based on a carefully designed finite element scheme for the direct scattering problem. The method has several merits: 1) the linear sampling method is used to quickly obtain a good initial guess; 2) finite Fourier series are used to approximate the boundary of the obstacle, which is decoupled from the boundary used by the finite element method; and 3) the mesh is fixed and hence the stiffness matrix, mass matrix, and right hand side are assembled once and only minor changes are made at each iteration. The effectiveness of the proposed method is demonstrated by numerical examples.