Inverse Conductive Media Scattering By A Class Of Permeable Obstacles

In this dissertation,we studied the theory and algorithm of conducting bound-ary inverse scattering problems with unknown buried objects,which are widely used in the fields of medicine and physics,such as the detection of underground mineral deposits（detection of oil,etc.）,the detection of internal damage of large objects,electromagnetic imaging,nuclear magnetic resonance（diagnosis of internal diseases of human body）,etc.For different buried objects（such as Neumann boundary conditions and Dirichlet boundary conditions）,we conducted the well-posedness of the positive scattering problem and the numerical inversion of the inverse problem and numerical experiments are conducted.First,for the Neumann scattering problem,we established the well-posedness of the positive problem by using the variational method,Reilich lemma,Riesz-Fredholm theorem,and Lax-Milgram lemma.For the inverse problem,based on the information of far-field operator,we constructed a series of operators,which can demon-strate the inversion and reconstruction of conductive boundary scatterers using the factorization methods.Then,for the Dirichlet scattering problem,by introducing a special Hilbert space and using variational method,we established the solution of the positive scattering problem.However,we can not apply the classical factorization method directly,since the middle operatorof the factorization of the far-field operatoris only compact.In this case,we have developed a modified factorization method by a sequence of perturbed operators₈)to achieve the inversion and reconstruction of conductive boundary scatterers.Next,based on the theoretical foundation of the previous theorem,we continued to consider the reconstruction of the model for the scattering problem of conductive media.Through numerical inversion experiments in two-dimensional space,we verified the effectiveness and practicality of the factorization method under Neumann and Dirichlet boundary conditions,and we extended to the impedance problem.Finally,we made a summary of the main ideas and research methods of this dissertation,and we will expect research on conductive scattering problems under impedance boundary condition or refractive index R（n）<1.And we will we will trust to develop two-dimensional models into three-dimensional models in numerical experiments.