Regularization Methods And Optimality Analysis For Inverse Boundary Value Problems

In this paper, from the viewpoint of optimality analysis, we consider three classical inverse boundary value problems: the inverse heat conduction problem, backward heat conduction problem and Cauchy problem for Laplace equation. They are all severely ill-posed problems, and The ill-posedness becomes sharp as the unknown solution is closer to the boundary point. Therefore, restoring stability of the solution, especially restoring stability of the solution on the boundary is very important for practice background and theoretical research. So far, many regularized methods only have the numerical results without error estimates, there are also some methods which can give the order optimal error estimate, but few work is denoted to the optimality, and the results on the boundary are less.In Chapter 2, we discuss optimality for the Cauchy problem for Laplace equation in unbounded strip region in the interval (0,1]. Meanwhile, we provide two regularization methods (i.e., the generalized Tikhonov regularization and the generalized singular value decomposition) and the choice rule of the regularization parameterα. When the regularization parameterαis chosen according to the rule, it can realize the optimal error bounds. Moreover, we give filtering method and Fourier regularization method about this problem. When the regularization parameters are chosen suitably, we obtain the order optimal error bound and logarithmic stable estimate between the exact solution and its approximation, respectively.In Chapter 3, respectively, we analyze optimality of error estimate for the Cauchy problem for Laplace equation and inverse heat conduction problem on the boundary. And we both obtain the optimal error bounds between the exact solutions and their regularized approximations under the logarithmic source conditions.In Chapter 4, by using the discretization regularization, we study Cauchy problem for Laplace equation and backward heat conduction problem in the bounded region. Here, we give the logarithmic stability error estimate between the exact solutions and their regularized approximations which are obtained by the spectral cutoff method, the least squares method and the dual least squares method.