Some Research On Regularization Method In Inverse Problems

As an important class of mathematical physical problems, inverse problems have developed into a popular research direction. Solving an inverse problem is to determine unknown causes based on observation of their effects. Nowadays, inverse problems have been used in many fields,such as inverse medium scattering, computerized tomography, etc, and the theory and methods are novel and challenging.Two essential difficulties appear frequently. One is the observation data possibly does not belong to the corresponding set to the exact solution, another is that the approximation is not stable. Thus inverse problems are often ill-posed, and most are non-linear. Regularization technique is an effective method of solving ill-posed inverse problems. Since regularized parameter influence the convergent rate, which becomes more and more important, and there are several significant results gotten by many researchers.Based on the merit of computing global minimum with homotopy continuation method in [6], a posterior regularized homotopy continuation method is presented in this thesis. Regularized parameter in the iteration is decided by Morozov discrepancy principle. We realized the computation in inverse medium scattering problems with multi-experimental data of far field pattern. Moreover, the analytical result of that the selected regularized parameter in each iteration accord with Tikhonov regularization theory from a few of numerical experiments is stated. The method of posterior regularized parameter can overcome the shortcomings of prior choose, and the numerical examples show the practicability of our new methods in inverse problems with perturbation data.