Theory Analysis And Numerical Implementations Of Factorization Method For Several Complex And Mixed Inverse Scattering Problems

The field of scattering theory has been one of a particularly active fields in Applied Mathematics and Physics.It is widely used in medical imaging,nondestructive testing.seismic exploration and other fields.The investigation of scattering theory can be divided into the direct and inverse problems.The direct problem is to study the well-posedness of a boundary value problem of the Helmholtz equation or Navier equation.And the inverse problem is to recover the location.shape.structure and physical properties of the scatterer from the far field pattern or other measurement data.In the practical applications.the scatterers are very complex.So in this paper.we consider different types of mixed scattering problems of acoustic waves.elastie waves and fluid-solid interaction scattering.The first chapter introduces the background of the scattering theory.The first section states the investigation background and significance of the scattering theory.The second section presents three types of scattering models,that is.acoustic scattering.elastic scattering and fluid-solid interaction scattering problems.The third section describes three theory frames of the factorization method in detailed.The fourth section states the main work and the present situation of our study.The fifth section shows the structural arrangement of this paper.The second chapter introduces the basic tools.important theorems and main methods which will be used in the scattering t heory.The fundamental solutions of the Helmholtz equation and Navier equation in two dimensional spaces and their far field patterns are described in the first section.The Sobolev spaces usually used in the scattering theory are presented in the second section.The potential theories and the related properties of the single-and double-layer boundary integral operators are stated in the third section.The common basic theories in scattering theory are presented in the fourth section.In the third chapter,we investigate the mixed acoustic scattering problem by an in-homogeneous medium with unknown buried objects and an impenetrable obstacle with Dirichlet boundary,condition.Firstly,we transform the original problem into an equiv-alent variational formulation by using the Dirichlet-to-Neumann mapping,and we apply the variational method to show the well-posedness of the direct,scattering problem.Then,the factorization method is used to solve the corresponding inverse scattering problem.Finally,some numerical experiments are presented to illustrate the feasibility and effi-ciency of the inverse algorithm.In the fourth chapter,we consider the mixed elastic scattering problem by an impen-etrable obstacle with Neumann boundary condition and a crack with Dirichlet bound-ary condition.Based on the Betti representation theory and the boundary condition,the well-posedness of the direct problem is proven by the boundary integral equation method.And we make use of the modified factorization method to deal with the inverse problem.Furthermore,the diferences between the rigorous mathematical theory and the numerical implementation are illustrated by the numerical experiments.In the fifth chapter,we study the mixed fluid-solid interaction scattering problem with a buried object inside.We solve the direct scattering problem by combining the boundary integral equation method and the variational techniques.To recover the in-teraction surface between the fluid and solid.we consider the factorization method.For the reconstruction of the imbedded obstacle.we show a mixed reciprocity relation be-tween the Green's function and the total field,and we also construct an identity relation between the unitary operator S and the interior displacement field.In the sixth chapter,we investigate the interior transmission eigenvalue problem by an anisotropic medium with partial coated boundary.Firstly,we show the well-posedness of the interior transmission problem by the variational method under special conditions.Then,the analytic Fredholm theorem is used to prove the discreteness and existence of the eigenvalues under the condition that the index refraction n equals to one.Finally.we consider the case that n is not always equal to one where the discreteness of the eigenvalues has been achieved by the T-coercive method under certain limitations.