Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm
Journal of Computational Physics
Journal of Computational Physics
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In this paper, a regularized homotopy continuation method is presented for numerical solution of nonlinear ill-posedness inverse problems. Two major difficulties for solving these inverse problems by an optimization method are the ill-posedness and the presence of many local minima. Classic iterative methods, such as Gauss--Newton or Levenberg--Marquardt algorithms, offer fast local convergence but might not be able to compute the global minimum. Based on a natural concept of multi-experimental data, a regularized homotopy continuation method is constructed to compute the global minimum. As the experimental index t increases continuously, the global minimizer can be computed continuously by using local optimization methods. By discretizing the continuous homotopy method, various recursive linearization algorithms are developed. These algorithms are applied to numerical solution of an inverse medium scattering problem which reconstructs the refractive index of an inhomogeneous medium from limited aperture measurements of the far field pattern of the scattered fields. It is assumed that the data is measured at multiple frequencies. The inverse scattering problem with limited aperture is challenging, since without full aperture measurements, the ill-posedness and nonlinearity of the inverse problem become more severe. An efficient regularized iterative linearization method (recursive linearization with respect to the wave number) is developed for solving the inverse problem. The convergence of the iterative method is examined by numerical examples.