Some new results on an inverse medium scattering problem

A two-dimensional (2D) inverse medium scattering problem (IMSP) for the Helmholtz equation is considered. The IMSP aims to reconstruct a 2D coefficient function representing the refractive index of the medium from measurements of the scattering data at a finite number of positions. To enhance the reconstruction accuracy, scattering data from multiple incident waves and at one low frequency and one high frequency are used. To solve the IMSP, a frequency-hopping least-squares minimization algorithm is used. The discrete adjoint equation method is used for computing the gradient of the discrete least-squares objective functional. In addition, an efficient method for solving the disretized Lippmann-Schwinger equation is also presented. Numerical examples demonstrate that the proposed method can reconstruct complicated shapes with high accuracy.