The First Kind Fredholm Integral Equation Of Multi-level Fast Algorithm

The Fredholm integral equations of the first kind arise naturally in science and technology, which are special cases of inverse problems. Most of these problems are ill-posed problems. In order to treat the ill-posed problems, regularization methods are often introduced, which convert the problems to related well-posed problems. When we use iteration methods to solve the regularization equations or use some discrepancy principles to determine the regularization parameter, repeated iterations have to be done, which involve a great quantity of computations. Therefore constructing a fast, stable and efficient algorithm is particularly important. In this paper, we consider numerically solving this problem by applying fast multiscale Galerkin method. The dissertation consists of four chapters.In chapter1we briefly introduce the ill-posed prolems, then state that the integral equations have been used in various fields, and classification of the integral equations; study of the first kind Fredholm integral equations methods, and summarizes the main work of this thesis.In chapter2we briefly describe the regularization theory, linear ill-posed problems of regularization methods, regularization parameter choice strategies and projection methods.In chapter3We develop multilevel Jacobi and Gauss-Seidel type iteration methods with compression technique for solving ill-posed integral equations. The method leads to fast solutions of discrete regularization methods for the equations. Then choice for a posteriori regularization parameter is proposed. An optimal convergence order for the method with the choices of parameters is established. Finally, numerical experiments are given to illustrate the efficiency of the method.In chapter4We develop multilevel Jacobi and Gauss-Seidel type iteration methods based on fast collocation methods for solving ill-posed integral equations. Firstly, we introduce the multilevel iteration methods for solving the discrete equations resulting from ill-posed integral equations of the first kind by using fast collocation methods. Then choice for a posteriori regularization parameter is proposed..