Numerical Methods For Two Kinds Of Inverse Problems On Partial Differential Equations

Numerical methods of inverse problems play an important role in modern science. In this thesis, we investigate the numerical methods for two kinds of inverse problems of partial differential equation, i.e., the identification of the corrosion boundary and Cauchy problem for Laplace equation and the identification of a Robin coefficient for heat conduction problem. They are ill-posed. In particular, a small perturbation in the data may result in an enormous deviation of the solution.The non-destructive evaluation of corrosion boundary becomes more and more im-portant in engineering and mathematics. In general, corrosion takes place on an inac-cessible part of boundary. Our task is to determine the shape of a corrosive boundary from the measured data. In Chapter 2. we use the method of fundamental solutions (MFS) to deal with the ill-posed problem. Since the resulting matrix equation is badly ill-conditioned. a regularized solution is obtained by employing the Tikhonov regulariza-tion (TR) technique, while the regularization parameter for the regularization method is provided by the generalized cross-validation criterion (GCV).The location of source points in MFS is a difficult point. In Chapter 2, the source points are uniformly distributed on the fictitious closed boundary. In Chapter 3, we first employ an adaptive technique to choose an arrangement of source points for ill-posed problem. Under the noisy data, the TR method is used, while the regularization parameter is chosen by the L-curve criterion (LC).For the examples with exact solution, the MFS can result in a good approximation, even if the data are insufficient. In Chapter 4, we apply the MFS to solve several numerical examples for Cauchy problem of elliptic operator. It verifies the result in paper [EABE,31(4):373-385,2007]. We point out the effectiveness of numerical method depends on the numerical example. For some example contains non-smooth boundary data. all of the MFS-related approaches in [EABE,31(4):373-385.2007] fail.The identification of a Robin coefficient for heat equation is a non-linear problem, i.e. identifying a Robin coefficient from boundary temperature measurement. In Chapter 5. we investigate the uniqueness in the weak sense. then transform the problem to a variational formulation. The conjugate gradient method (CGM) is used to solve the optimization problem. In the process of iteration. the CGM involves the solution of three problems. i.e., the direct. sensitivity and adjoint problems. The key point is to obtain the gradient of the objective functional. We apply two kinds of method:the definition and Lagrange-nmltiplier method. Under the noisy data, the CGM has typical 'semiconvergence' phenomenon. We take discrepancy principle as a stopping criterion. Then the stopping number plays a role as a regularization parameter. And we investigate the effect of regularization parameter in Tikhonov functional on the numerical results.