Numerical Study On The Scattering And Inverse Scattering Problems For Open Arcs In The Time Domain

Wave is a very important tool for the transmission of information.Wave scattering has always been an important research subject in applied mathematics.Unlike the study of scattering and inverse scattering in the frequency domain,the theory and algorithm of scattering and inverse scattering in the time domain are not very mature,but they are much closer to practical applications.This thesis aims to study the open arc scattering and inverse scattering which are widely used for geophysical prospecting and non-destructive testing in the time domain.My research work is mainly on:1.We study the forward scattering problem of two-dimensional open arc in the time domain.First,we give the well-posedness of the sound-soft open arc scattering problem in the time domain.Then we discuss the numerical method for solving the retarded potential boundary integral equation.Finally,we give several numerical examples of the forward scattering problem to demonstrate the feasibility and effectiveness of the proposed method.2.We study the inverse scattering problem of two-dimensional open arc in the time domain.We give a proof of the blow-up property of the time domain linear sampling method for the case of sound-soft open arc.Then we give several numerical examples of time domain linear sampling method.We also discuss the time domain direct sampling method.With this method,we reconstruct single and multiple arcs with different boundary conditions,and study various factors that affect the reconstruction results.3.We propose other two methods:the Newton iterative method in the Laplace domain and the nonlinear optimization method.The former needs to transform the scattered data with scaled DFT,and carry out the iterative process in Laplace domain.The latter constructs the objective functional with the time domain data directly,and uses the lsqnonlin in Matlab to find the open arc by minimizing the objective functional in the finite dimensional function space.Finally,we give several numerical examples to demonstrate the feasibility and effectiveness of the nonlinear optimization method.