Reciprocity Gap Method For An Interior Inverse Scattering Problem For Cavities

In recent years,with the wide application of inverse scattering of acoustic and electromagnetic wave in real life,as well as the nonlinearity and ill-posed of inverse scattering problem,inverse scattering of acoustic and electromagnetic wave problem has attracted the attention and research of many scholars at home and abroad,and has also become a hot research topic in the inverse problem of mathematical physics equations.At the same time,in the field of nondestructive detection,such as nuclear power production,need to accurately determine the position and shape of the cavity,therefore,this paper mainly uses the reciprocity gap method to study the interior inverse scattering problem for cavities.The paper is divided into five chapters,the specific content is as follows.In chapter 1,we mainly introduce the research background and significance,research status and innovation of this paper.In chapter 2,we introduce the relevant concepts and theories involved in this paper.In chapter 3,we study the inverse scattering problem of an interior impenetrable cavity with Neumann boundary.Firstly,it is proved that the location and shape of the interior impenetrable cavity with Neumann boundary condition can be uniquely determined by the measured data of the interior point source by using the reciprocity relation and the well-posed of the Helmholtz equation.Secondly,according to the linear sampling method,the reciprocity gap method is introduced,and a single-layer potential energy is introduced,combining with the potential energy theory,Green's formula and the convergence of the sequence in the Sobolev space,the determination method and corresponding steps of the cavity boundary point are obtained.Finally,the numerical simulation shows that the boundary of cavity inverted by this method is clearer.In chapter 4,we study the inverse scattering problem of a partially coated mixed boundary with an interior penetrating cavity.Firstly,Using Green first theorem and LaxMilgram lemma,the existence and uniqueness theorem and boundary uniqueness theorem of the solution of this problem are proved under the external transmission boundary.Secondly,by using the properties of dual operator and the continuation theorem,the density of measured data on the cavity boundary is obtained.And then,the singlelayer potential energy is introduced,it is proved that the data from the point source correspond to the data from the measured curve by using the method of reciprocity gap and single-layer potential energy theory.Finally,through the method of convergence of basic sequence in complete space,the step theorem for reconstructing the boundary of a cavity is obtained.In chapter 5,we summarize the research of this thesis and the prospect of the further research.