Studies On The Regularization Methods For Some Inverse Problems

Inverse problems are usually ill-posed, and small change in the given data may cause dramatically large change in the solution. Thus in the study of inverse prob-lems, we are most concerned about the restoration of the stability of the solution. In order to solve this problem, mathematicians introduced various kinds of regu-larization methods, such as Tikhonov regularization method, Landweber’s iteration method and so on. In recent years, some non-classical regularization methods ap-peared in the research of inverse problems, such as Fourier truncation method and wavelet method, etc. In this thesis, we use two non-classical regularization methods to systematically study several kinds of inverse problems, and obtain convergence analysis in theory. Furthermore, the algorithm which is applied to compute the numerical solutions is also presented.This thesis includes four chapters. The first chapter briefly introduces inverse problems and the regularization theory for ill-posed problems.The second chapter studies one kind of non-characteristic Cauchy problem. We firstly transform the problem into the second kind of integral equation, and three methods are used to construct its approximate problem:the integral equation method, Fourier truncation method and modified integral equation method. We prove the well-posedness of the three constructed approximate problems, i.e., ex-istence, uniqueness and stability of the solution. Finally, the convergence analysis between exact solution and its approximation is given for the suggested methods.The third chapter considers the application of wavelet method in dealing with three kinds of inverse problems. First of all, for the problem of backward heat e-quation, we use wavelet projection method to project the given data on the wavelet space Vj and compute the approximate solution from the projection data, the sta-bility analysis of approximate solution is obtained. Afterwards we discuss the time fractional inverse diffusion problem and space fractional backward diffusion problem, respectively. As the wavelet projection method only project the data, the approx-imate solution can not be ensured to belong to the wavelet space Vj. In order to overcome this deficiency, we combine wavelet method with Galerkin method, i.e., we solve the latter two problems by using wavelet-Galerkin method so that the solution is restricted on wavelet space Vj, and we obtain the convergence analysis between the exact solution and its approximation. Iii the fourth chapter, we give the numerical implementation for the proposed methods. For the non-characteristic Cauchy problem, we adopt a line method and the problem is modeled as an initial value problem for ODEs, which will be regu-larized and discretized. For the wavelet projection method, the solving process is relatively simple because the analytical expression of solution can be obtained. We only project and filter the data, and then solve problem by using the expression of the solution. While for the latter two equations, we directly solve the finite-dimensional approximation of the infinite-dimensional ordinary differential equations which are obtained by Galerkin method, where the matrix Dj is derived by applying wavelet transform to the difference approximate matrix of time fractional derivative or space fractional derivative. The numerical results demonstrate that our proposed methods give a good approximation of the original problem.