Multi-scale Compressed Projection Accelerated Iterative Algorithm For Solving Linear Integral Equations

This paper mainly discusses the regularization of Landweber iteration based on Nesterov acceleration scheme in Hilbert space,the implicit nonstationary iteration in Hilbert space and the nonstationary iteration Lavrentiev regularization in Banach space.the multi-scale compressed projection method is used to solve these linear ill posed problems in finite dimensional space.The relatively perfect theoretical knowledge is established,and the corresponding posterior parameter selection strategy and numerical algorithm are given.In the second chapter,we consider the Landweber iterative method to solve the problem after the acceleration of the Nesterov scheme.The problem is discretized using the Galerkin projection method with compression strategy,the convergence and error estimates of the approximate solution are given,the iteration stop criterion is used to terminate the iteration,and a discrete Nesterov accelerated iterative algorithm is proposed to make the approximate solution reach The optimal convergence rate O(?(2?/(2?+1))).In the third chapter,we discussed the implicit nonstationary implicit iterative regularization method for solving linear integral equations in the Hilbert space,and proved the prior error estimation formula of the approximate solution.Then the Morozov deviation principle is used as the posterior parameter iteration stop criterion.The results show that this method ensures that the approximate solution reaches the optimal convergence rate O(?(2?/(2?+1))).and the accuracy of the theoretical results is further verified by numerical experiments.In the fourth chapter,We consider extending the iterative regularization of nonstationary Lavrentiev to the Banach space for research.The multi-scale collocation method in the Banach space and related theories are introduced.The nonstationary Lavrentiev iterative equation is compressed and projected,and a multiscale collocation method for quickly solving integral equations is obtained.Similar to Chapter 3,a detailed proof process of error estimation and convergence is given,and the iteration is terminated using the Balance Principle to ensure that the approximate solution reaches the optimal convergence rate O(?(?/(?+1))).