Solving The Fredholm Integral Equation Of The First Kind With Perturbed Initial Data Base On Truncation Strategy

In this thesis we develop a fast multiscale method solving the first Fredholm integral equation with notexactly given input data. However, the first kind Fredholm integral equation is a typical ill-posed problem. Inorder to treat the Fredholm integral equation of the first kind, regularization methods are often introduced, whichconvert the problems to a Fredholm integral equation of the second kind. Based on the multiscale method, wepropose a matrix truncation strategy and the convergence rate of the Tikhonov regularization is achieved byusing a posteriori parameter choice strategy. Up to present time, research results are very few about integralkernel of the first Fredholm integral equation with not exactly given input data wich is the focus of our research.The dissertation consists of three chapters.In chapter1we briefly introduce the historical background and the recent development of the first kind ofFredholm integral equations，regularization methods, regularization parameter choice strategies and outline themain work of this thesis.In chapter2we develop multilevel Galerkin methods with truncation technique for solving Fredholm integralequations of the first kind with not exactly given input data. Firstly, we introduce the multilevel Galerkin methodsfor solving the discrete equations resulting from ill-posed integral equations of the first kind. Then the choicefor a posteriori regularization parameter is proposed. An optimal convergence order for the method with thechoices of parameters is established. Finally, numerical experiments are given to illustrate the efficiency of themethod.In chapter3we develop a multilevel augmentation method for solving Fredholm integral equations of thefirst kind with not exactly given input data. Firstly, we introduce the multilevel collocation methods for solvingthe discrete equations resulting from ill-posed integral equations of the first kind, and then solving the discreteregularization equation by multilevel augmentation algorithm. Then finally, choice for a posteriori regularizationparameter is proposed and an optimal convergence order for the method with the choices of parameters isestablished.