Some Research On Numerical Methods Of Inverse Obstacle Scattering Problems

This thesis discusses several numerical methods for solving the inverse obstacle scattering problems.The uniqueness of these problems is also established.Numeri-cal examples are provided to demonstrate the accuracy and stability of the methods.The first chapter of this thesis is the introduction.In this part,we introduce the physical background of the inverse obstacle scattering problems and the current developments in this field.This chapter also presents preliminary and fundamental knowledge of inverse scattering theory such as the Helmholtz equation and the theory of layer potentials.In the second chapter,we introduce a direct imaging method for the inverse obstacle scat-tering problem,which applies to both the sound-soft and sound-hard boundaries.We first use the Fourier-Hankel expansion to approximate the scattered field and then pro-pose the indicator functions according to the boundary conditions.It can be proved that the indicator functions vanish only on the boundary of the obstacle.The numerical re-sults of the reconstruction are presented to demonstrate the effectiveness and viability of the proposed method.In the third chapter,we consider the co-inversion problem of determining the shape of the obstacle and the location of incident point sources.The single-layer potentials are used to decouple the total field data into the approximate in-cident and scattering fields.Then we use the optimization method to reconstruct the shape of the obstacle and analyze the stability of the optimization method.In addition,we employ the direct sampling method to determine the location of the point sources.We also provide some numerical examples to show the effectiveness and viability of the proposed method.In the fourth chapter,we investigate the co-inversion problem of simultaneously determining the shape of the obstacle and the location of its excita-tion sources from phaseless near-field data.We decouple the problem by adding some artificial point sources.The uniqueness of the obstacle boundary and the source loca-tions are proved respectively.Several numerical examples are presented to validate the theoretical findings.The last chapter is the conclusion of this thesis.