Numerical Solutions For The Cauchy Problem For The Helmholtz Equation And Its Application

We consider the Cauchy problem for the Helmholtz equation, which is a prob-lem of reconstructing the Cauchy data on the interior boundary from Cauchy data given on the exterior boundary. Using Green’s representation theorem, we transform the Cauchy problem into a system of two boundary integral equations. Noticing the ill-posedness of this system, regularization technique has to be used to solve this problem. In this paper, we first use the classical Tikhonov regularization technique to deal with this problem. Then, based on the singularity features of the integral equations, a quasi-Tikhonov regularization technique is proposed. The advantages of this method are as follows:on one hand, we can obtain simultaneously the Dirich-let data and Neumann data by solving a regularized system; on the other hand, compared with the method based on the classical Tikhonov regularization technique for solving the equations, the proposed method is more simple from the numerical point of view, and needs relatively small amount of computations in numerical im-plementations. Some numerical results, with exact and noise measurement data, are also presented to show the efficiency and accuracy of our scheme. As an application of our method for the Cauchy problem, we consider the problem of reconstructing the boundary impedance from scattering data. We first solve the direct scatter-ing problem to get the Cauchy date on the exterior boundary, and then use the quasi Tikhonov regularization method to calculate the Cauchy data on the interior boundary. Finally, combined with some technique of regularization to reconstruct the boundary impedance coefficient.Our thesis consists of four chapters.In the first chapter, we introduce the mathematical model of the Cauchy prob-lem and reconstruction of the boundary impedance coefficient. Then we give some well-known results and describe our main works.In chapter two, we give some preliminaries, containing potential theory and regularization theory.In chapter three, we study the construction of the Cauchy data on the interior boundary. We transform the Cauchy problem into a system of two boundary inte-gral equations, which is proved to be uniquely solvable. Then, we use the classical Tikhonov regularization method and our new quasi Tikhonov regularization to solve this equations.In chapter four, we study the reconstruction of the boundary impedance co-efficient. Use our quasi Tikhonov regularization, we first reconstruct the Cauchy data in the interior boundary from measured data given on an accessible boundary. Then, combined with Tikhonov regularization technique, we reconstruct the bound-ary impedance coefficient. Several numerical examples are presented to show the efficiency and validity of the reconstruction scheme.