The Method Of Fundamental Solution For Inverse Problems Of Laplace’s Equation

The Cauchy problem for Laplace’s equation is a severely ill-posedness of the prob-lem and arises from many branches of science and engineering such as non-destructive testing, steady-state inverse heat conduction, and electro-cardiology. In this thesis, we mainly discuss convergence analysis of the method of fundamental solution (MFS) for the Cauchy problem of Laplace’s equation and apply the MFS and Tikhonov regularization to solve an inverse impedance problem for Laplace’s equation.In the second chapter and the third chapter, we discuss the Cauchy problem of Laplace’s equation in a simply-connected bounded domain and an annular domain, re-spectively. We propose a variational-type method of fundamental solution for solving the Cauchy problem of Laplace’s equation and obtain the stable regularization solutions in a simply-connected bounded domain and an annular domain, respectively. Based on a global conditional stability and a conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a reg-ularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.In the fourth chapter, we consider an inverse impedance problem for Laplace’s equa-tion. For the inverse impedance problem of Laplace’s equation, we first transform the problem into an optimization problem based on the MFS and Tikhonov regularization and then use the popular conjugate gradient method to solve the minimization problem. We compare several stopping rules in the iteration procedure and try to find an accurate and stable approximation. At last, Numerical results will show the effectiveness of the proposed method.