Decomposition Method For An Interior Inverse Scattering Problem For Cavities

Acoustic inverse scattering problem plays an important role and possesses a widely promising application in many fields,such as radar,sonar,medical imaging,nondestructive test,geological exploration etc,especially in mathematical physics.mostly have considered the external inverse scattering problem in previous studies,In this paper,we consider an internal acoustic inverse scattering problem.But there are two difficulties we have to face in the problem of internal acoustic inverse scattering,one is ill-posed,the other is nonlinear.Therefore,it is more difficult to solve the problem,the decomposition method is used to solve the internal inverse scattering problem in this paper.This is mainly studies two aspects with decomposition method.One is the inverse scattering problem of impenetrable cavity under the condition of boundary theory,another is the inverse scattering problem of penetrable cavity with transmission boundary condition.This paper contains the following 5 chapters:Chapter 1:This chapter mainly sketches the background,significance and actuality of inverse scattering problem,lists several relatively mature existing methods,and briefly analyzes the advantages and disadvantages of these methods.As well as the brief introduction of main work in this paper.Chapter 2:This chapter mainly introduces the basic knowledge involved in this paper,including the relevant theoretical knowledge of Helmholtz equation,potential theory,the concept of ill-posed problem and regularization method to solve this problem and so on.Chapter 3:We reconstruct the shape of an impenetrable cavity with Heumannboundary condition in two-dimensional space by using a single point source.Firstly,mathematics model of acoustic inverse scattering problem should be established.Secondly,we present the proof of the uniqueness theorem.Then,we discuss the decomposition method for the inverse problem which breaks two steps,the first steps deals with the ill-posed,and then deals with the nonlinearity.Lastly,several numerical examples to illustrate the feasibility and effectiveness of the method.At the same time,the process of reconstructing the cavity by decomposition method with multiple points is given.Chapter 4:The decomposition method is extended to the case of penetrable cavity with transmission boundary condition.As same as the third chapter,a single point source is used to invert the scatterer boundary,mathematics model of acoustic inverse problem from penetrable cavity is established,as well as the existence and uniqueness of inverse scattering problem is proved.Then acoustic inverse scattering problem will be solved by two steps according to the thought of decomposition method.Lastly,the boundary of cavity is reconstructed,the process of solving multiple points sources is given as well.Chapter 5:This chapter summarizes the main idea of the whole paper and shows more directions of further studies.