Studies On Direct Sampling Methods For Inverse Point Source Problems

Partial differential equations(PDEs)are usually used to model various complex realworld phenomena in different fields of science and engineering.In recent years,there has been an increasing interest in inverse problems of determining unknown parameters in PDEs such as the source terms or boundary data.Particularly,many theoretical and numerical approaches have been the task of various investigations for the inverse source problems of wave equations.The goal of the inverse point source problems is the identification of the parameters of the unknown point source(e.g.location and moment)from measurement data(e.g.near-field or far-field data).The inverse point source problems are mathematically challenging because they are usually ill-posed in the sense of lacking uniqueness or stability.Therefore,this thesis is devoted to the inverse point source problems of acoustic,electromagnetic and elastic waves with a fixed frequency.The purpose of this thesis is to design reconstruction algorithms for the identification of the parameters of the target sources from the measured data with a fixed frequency.To this end,the direct sampling methods are used.The key novelty is to adopt some new indicator functions to recover the locations and moments or intensities of the point sources.These proposed methods are simple and easy to implement,computationally cheap and tolerant to the measurement perturbations.The first chapter introduces some preliminary materials related to the inverse problems.We first give a brief description of the concept of direct and inverse problems,together with the ill-posedness and regularization methods for inverse problems.Then,we provide some recent developments of the inverse scattering and inverse source problems.An important part of this chapter is the description of the basic idea and some recent works on the direct sampling methods for solving the inverse scattering and inverse source problems.In addition,this chapter describes the research objectives and the organization of the thesis.The major contributions of our present research consist of three chapters:In chapter 2,we are concerned with the direct sampling method for solving the inverse problems of identifying point sources for the Helmholtz equation from far-field data.We develop some novel indicator functions which could determine the locations and intensities of the target multipolar sources with a single wavenumber.Theoretically,the indicating behavior is rigorously analyzed and the stability estimate is established.Two and three-dimensional numerical experiments are conducted to illustrate the effectiveness and robustness of the proposed approach.The numerical results show that the proposed method has the capability of recovering the locations and intensities of the point sources.Meanwhile,it is simple,totally direct and computationally cheap with only inner products involved.Moreover,it highly tolerates to the measurement noise.Chapter 3 is dedicated to the study of the inverse source problem for time-harmonic Maxwell's equations.We extend the direct sampling scheme from the acoustic case to the electromagnetic case and develop a direct sampling method to recover the location and the moments of electric current dipoles from near-and far-field data at a fixed frequency.The method is based on a new indicator function and requires the evaluation of simple integrals only.It is easily implementable and efficient.We established the stability results.Numerical examples demonstrate the robustness of the method with respect to noisy measurements and its capability in identifying the source locations and the moments.The inverse elastic source problem is the content of chapter 4.In this chapter,we consider an extension of the direct sampling method using acoustic wave(Helmholtz equation)into elastic wave(Navier equation).Because of the coexistence of compressional and shear waves that propagate at different speeds,the problem of elastic wave takes a more complicated form than the acoustic case.We propose a non-iterative sampling technique for identifying locations and moments of the moment tensor point sources from the far-field data.The proposed direct sampling method is based on a novel indicator function and the indicating behaviors of this function is analyzed.The major advantage of this method is the ability to recover the locations and moments only through computing the integrals of the far-field data and an exponential function at each sampling point.Theoretically,we obtained the stability estimates.Numerical experiments are conducted to validate the theoretical justifications and demonstrate the effectiveness and robustness of the proposed methods.