Well-posedness Analysis In Scattering And Inverse Scattering Problems

Electromagnetic scattering problem has played a central role in 20 th century mathematical physics.In recent years,theoretical research,numerical calculation and analysis methods for scattering problems have been widely used in many fields such as scientific frontiers and major national strategic needs.In this paper,two kinds of scattering problems are studied,one is scattering and inverse scattering of obstacles in an unbounded structure.We have made a well-posedness analysis and obtained some results in uniqueness and stability.The other is a kind of scattering problem in near-field optics.By discretizing the integral equation into a linear system with special structure,we propose a effective numerical method.This paper is organized as follows:In chapter 2,we introduce some preliminary knowledge,including Maxwell's equations in electromagnetics theory,Helmholtz's equations,the properties of the solutions and boundary conditions.We also introduce the Fredholm alternative and the unique continuation theorem,which play a key role in the well-posedness analysis.The understanding of this part lays the foundation for the next two chapters.In Chapter 3,we focus on the rigorous analysis of the direct and the inverse obstacle scattering problem in unbounded structures.We introduce an integral radiation condition for the direct scattering problem.The asymptotic behaviour of Green's function is presented.An equivalent system of boundary integral equations is proposed.Based on Fredholm alternative theorem,the uniqueness of the solution is established for the direct scattering problem.For the inverse problem,we prove that the obstacle and the infinite rough surface are uniquely determined by the reflected and transmitted wave fields measured on the plane surfaces which are above and below the infinite rough surface,respectively.The proof is based on a combination of the Holmgren uniqueness and unique continuation.Based on the well-posedness arguments for the direct scattering problem,we obtain the domain derivative of the wave field with respect to the change of the shapes of the obstacle and the infinite rough surface.Moreover,a local stability is established.It indicates that the Hausdorff distance of two domains,which are characterized by two different obstacles and infinite rough surfaces,is bounded by the distance of corresponding wave fields if the two obstacles and infinite rough surfaces are close enough.In Chapter 4,we consider a class of scattering problems that arise in near-field optics.We focus on the numerical computation of the scattering problems with wave fields containing evanescent components.Based on the asymptotic expansion of special functions,the integral equation is discretized as a linear system with special structure,which is convenient for numerical computation.Physical characteristics of the interaction between the probe and the near field are investigated by using numerical calculations.Numerical experiments are included to demonstrate the effectiveness of the proposed method.