Regularization Methods For An Inverse Problem Of The Helmholtz Equation

The inverse problem appears in many areas of science and engineering, such as geophysics, medical imaging, non-destructive testing, and so on. This thesis main-ly discusses an inverse problem of the Helmholtz equation-the Cauchy problem for the Helmholtz equation. The Cauchy problem for the Helmholtz equation is severely ill-posed in the sense of Hadamard, especially the solution of the Cauchy problem is unstable, i.e. a small perturbation in the given Cauchy data will cause a large change to the solution, which makes the numerical computation very difficult. To overcome such difficulty, the regularized technique is applied to solve this kind of ill-posed problem. This thesis includes four chapters.The first chapter of this thesis firstly briefly introduces the theory about the inverse probelm and the regularization method, and then introduces the research background and status of the Cauchy problem for the Helmholtz equation, and finally briefly intro-duces the main work of this thesis.The second chapter mainly discusses the Cauchy problem for the Helmholtz equa-tion with homogeneous Dirichlet boundary conditions in a rectangular domain. In order to obtain a stable approximate solution of this problem, we use a filter regularization method to solve it, give the error estimates between the regularized and exact solutions under an a-priori parameter choice rule, and present two numerical examples. Both the theoretical and numerical results show that the filter regularization method used in this chapter is feasible and effective.The third chapter mainly discusses the Cauchy problem for the Helmholtz equa-tion with homogeneous Robin boundary conditions in a rectangular domain. We use a modified Tikhonov regularization method to deal with it. Under an additional a-priori bound assumption for the solution, we give the stable convergence analysis based on an a-posteriori parameter choice rule. Finally, the numerical results show that the modified Tikhonov regularization method applied in this chapter is feasible and effective.In the fourth chapter, the work on this thesis is summarized, meanwhile make the prospects for the future work.