DIRECT SAMPLING METHODS FOR INVERSE SCATTERING PROBLEMS

Recently, direct sampling methods became popular for solving inverse scattering problems to estimate the shape of the scattering object. They provide a simple tool to directly reconstruct the shape of the unknown scatterer. These methods are based on choosing an appropriate indicator function f on Rd, d=2 or 3, such that f(z) decides whether z lies inside or outside the scatterer. Consequently, we can determine the location and the shape of the unknown scatterer. In this thesis, we first present some sampling methods for shape reconstruction in inverse scattering problems. These methods, which are described in Chapter 1, include Multiple Signal Classification (MUSIC) by Devaney, the Linear Sampling Method (LSM) by Colton and Kirsch, the Factorization Method by Kirsch, and the Direct Sampling Method by Ito et al. In Chapter 2, we introduce some direct sampling methods, including Orthogonality Sampling by Potthast and a direct sampling method using far field measurements for shape reconstruction by Liu. In Chapter 3, we generalize Liu\u27s method for shape reconstruction in inverse electromagnetic scattering problems. The method applies in an inhomogeneous isotropic medium in R3 and uses the far field measurements. We study the behavior of the new indicator for the sampling points both outside and inside the scatterer. In Chapter 4, we propose a new sampling method for multifrequency inverse source problem for time-harmonic acoustics using a finite set of far field data. We study the theoretical foundation of the proposed sampling method, and present some numerical experiments to demonstrate the feasibility and effectiveness of the method. Final conclusions of this thesis are summarized in Chapter 5. Recommendations for possible future works are also given in this chapter

The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure

In this paper, we consider the inverse problem of recovering a doubly periodic Lipschitz structure through the measurement of the scattered field above the structure produced by point sources lying above the structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure is a perfect conductor partially coated with a dielectric. A periodic version of the linear sampling method is developed to reconstruct the doubly periodic structure using the near field data. In this case, the far field equation defined on the unit ball of R^3 is replaced by the near field equation which is a linear integral equation of the first kind defined on a plane above the periodic surface.Comment: 16 pages, Submitted for publicatio

A Two-stage Method for Inverse Medium Scattering

We present a novel numerical method to the time-harmonic inverse medium scattering problem of recovering the refractive index from near-field scattered data. The approach consists of two stages, one pruning step of detecting the scatterer support, and one resolution enhancing step with mixed regularization. The first step is strictly direct and of sampling type, and faithfully detects the scatterer support. The second step is an innovative application of nonsmooth mixed regularization, and it accurately resolves the scatterer sizes as well as intensities. The model is efficiently solved by a semi-smooth Newton-type method. Numerical results for two- and three-dimensional examples indicate that the approach is accurate, computationally efficient, and robust with respect to data noise.Comment: 18 pages, 5 figure