Iterative methods in inverse scattering problem

Abstract

Inverse scattering problems are to determine the property of the scatterer from the measured field outside the scatterer. These inverse problems arise in a variety of applications such as geophysical exploration, remote sensing, non-destructive evaluation, medical imaging and radar target recognition and so on. There have two difficulties to solve inverse scattering problems: nonlinearity and illposedness. For many inverse scattering problems, it suffices to derive approximate solutions when the scattering from inhomogeneity is weak. Most widely known and used methods are linearizations such as the Born approximation and the Rytov approximation, which are methods to overcome nonlinearity. Numerical methods that can overcome illposedness are the so-called regularization methods such as Tikhonov regularization, Landweber iteration, accelerated Landweber method and ν-method and so on. In this thesis, the mathematical background on inverse medium scattering problem is described. Physical background deriving Helmholtz equation in acoustic wave, Lippmann-Schwinger equation, far field pattern, remarks on solving inverse scattering problem, regularization scheme are summarized in some detail. Specially, the Born approximation is in brief described theoretically and two extended Born approximations suggested by Murch[40] and Jun & Choi[33] are presented in detail. Also, the numerical experiments on two extended Born approximations are presented and compared. Finally, the Born iterative method(BIM) and the distorted Born iterative method(DBIM) are formulated and analyzed rigorously. The sufficient conditions for convergence of the BIM and the DBIM are proposed.