| In image processing, signal processing, remote sensing technology, pattern recognition, and many other areas of science and technology are put forward in the inverse problem, which known as"mathematical physics problems". And solving these problems can be ultimately comes down to solve the first Fredholm integral equation. The inverse problem of this kind of problem is special, most of that have not qualitative. For ill-posed problems, we must adopt regularization method to get the approximate solution of the meaningful.Such as Tikhonov regularization method, Lavrentiev regularization method, landweber iterative regularization. But regularization method in the selection of regularization parameters is critical. The thesis mainly studies solve ill-posed integral equation multiscale algorithm. The full text is divided into three chapters.The first chapter, we briefly introduce the research status of the first kind Fredholm integral equation and the main work of this paper.The second chapter, we develop a fast multiscal Galerkin method for the first kind Fredholm integral equation with not exactly given input data via Tikhonov regularization.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.The third chapter, we develop Galerkin methods with compression technique for solving the first kind Fredholm integral equation. Firstly, multilevel augmentation methods with compression technique are developed for solving the first kind Fredholm integral equation, which can get stable solution; Secondly, we propose iterative regularization method which can improve the convergence rate of the solution.Then choice for a posteriori regularization parameter is proposed... |
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