Studies On Some Inverse Problems For The Helmholtz Equation

The Helmholtz equation arises in many practical physical and mechanical ap-plications, such as acoustic and electromagnetic wave propagation and scattering, vibration of structures and so on.In this thesis,we mainly discuss two kinds of ill-posed problems for the Helmholtz equation:the Cauchy problem for the Helmholtz equation and the inverse scattering problem for cavities with impedance boundary condition.In the second chapter and the third chapter,we discuss the Cauchy problem for the Helmholtz equation in a rectangular domain and a general domain,re-spectively. We use three regularization methods:the modified quasi-reversibility method,the modified Tikhonov regularization method and the truncation method to solve the Cauchy problem in the rectangular domain and obtain its stable reg-ularization solution; For the Cauchy problem in the general domain, by Green's formulation,we first transform it into a moment problem and then propose two regularization methods:the Talenti's method and the Tikhonov regularization method to solve a finite Hausdorff moment problem, and then obtain a stable regularization solution for the Cauchy problem. We give the stable convergence analysis for this problem in both domains and the numerical results show that our proposed methods are stable and effective.In the fourth chapter, we consider an inverse scattering problem for cavities with impedance boundary condition.It is shown that both the shape of the cavity and the surface impedance are uniquely determined by the measured data on a curve which locates in the interior of the cavity. We determine both the shape of the cavity and the surface impedance from the measured data by using the linear sampling method and the theory of the integral equation and the numerical results show our proposed numerical algorithms work effectively.