High Efficient Spectral Method For Eigenvalue Problem In Inverse Scattering

The acoustic wave transmission eigenvalue problem and Steklov eigenvalue problem in inverse scattering play an important role in inverse scattering theory.The acoustic wave transmission eigenvalue problem appears in inverse scattering theory of inhomogeneous media and has important physical background and wide applications.Since the transmission eigenvalue can be determined by far-field data,it can be used to estimate the properties of scattering material.Steklov eigenvalue problem is of great importance in the study of eigenvalue problems.In recent years,a non-selfadjoint Steklov eigenvalue problem has appeared in the inverse scattering theory and has attracted the attention of many scholars.These two problems are both non-selfadjoint and indefinite,and the fundamental theory of the usual elliptic eigenvalue problem can not be directly used to study it,which leads to great difficulty to the theoretical analysis and numerical calculation.This paper mainly focus on the high efficient spectral method of eigenvalue problem in inverse scattering.The main contents are divided into the following five Chapters:In Chapter 1,we introduce the background and significance of the research problems,summarize previous works,and give the research content and structure of the paper.In Chapter 2,we introduce some preliminary knowledge,mainly including some usual symbols which will be used in this paper,the basic properties of Jacobi polynomials,the weighted Sobolev spaces and the approximation properties of some projection operators.In Chapter 3,we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for the acoustic wave transmission eigenvalue problem.Firstly,we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem,and then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Possion equation.Secondly,based on polar coordinate transformation,we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems.Thirdly,we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition,and further establish the weak form and the corresponding discrete form.In addition,by utilizing the spectral theory of compact operators,we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system.Finally,we provide ample numerical experiments,and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.In Chapter 4,a spectral method for the Steklov eigenvalue problem in inverse scattering is proposed.Firstly,we establish the weak form and corresponding discrete scheme by introducing an appropriate Sobolev space and an approximation space,respectively.Secondly,according to Fredholm theory,the corresponding operator forms of the weak formulation and discrete formulation are derived.Then,the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators.In addition,a set of appropriate basis functions are constructed in the approximation space,the matrix form based on tensor product of the discrete scheme is deduced.The algorithm is extended to the case of circular domain.Finally,we present plenty of numerical examples,and the numerical results illustrate the effectiveness and high accuracy of the algorithm.In Chapter 5,we develop an efficient spectral approximation for the interior transmission eigenvalue problem in a circular/spherical domain.Firstly,we define a product type Sobolev space,and construct the corresponding approximation space by using a class of orthogonal polynomials in the unit ball.Then,by introducing an auxiliary function,we transform the original problem into an equivalent fourth-order mixed scheme,and derive the variational form and discrete form of the fourth-order mixed scheme.Moreover,we prove the error estimation of the approximate solution by using the approximation property of the orthogonal projection operator and Babu?ka-Osborn theory.In addition,we describe an implementation process of our algorithm in detail.Finally,we also present some numerical examples,and the numerical results indicate the effectiveness and high accuracy of the algorithm.