The Multiscale Collocation Method With Matrix Compression For Solving Integral Equation

In this thesis,we investigate the multiscale collocation method with matrix compression for solving the Fredholm integral equation of the first kind.The main body of the thesis is divided into two parts:in the first part,we adopt the fast multiscale collocation method with matrix compression strategy to discrete and solve the Fredholm integral equation of the first kind;in the second part,we combine multiscale collocation method with rmultilevel iteration method which is solve the equation of Lavrentiev regularization,the paper is organized as follows:The first chapter,we briefly introduction the concept of the Fredholm integral equation of the first kind and its relation with integral equation,ill-posed problem,inverse problem,then giving some preliminary knowledge to carry out the work of this thesis;next,according to the order of time,we list the typical domestic and foreign literatures for solving the Fredholm integral equation of the first kind;finally,the main work of this thesis is described.The second chapter,we adopt the fast multiscale collocation method to solve the first kind of Fredholm integral equation with sectorial operator in Banach space,we have generalized the work of our predeces-sors:the fast multiscale collocation method with matrix compression strategy for solving alternate iterative equations is presented,and it reduces the coefficient matrix computation of non-zero elements;according to the Balance principle,and posterior iteration stopping criterion is given to ensure that the regular solution is quasi-optimal.The third chapter,we present multilevel iteration method for solving the first kind of Fredholm integral equation:using multiscale basement discrete Lavrentiev regularization equation with compact support and vanishing moment is obtained,so that the linear system can be obtained with the hierarchy and numerical sparsity;then,multilevel iteration method is constructed by using high-,low-frequency decomposition tech-nique,and the prior error estimation is given,the parameter selection strategy is obtained through Balance principle,which made the regular solution is quasi-optimal.The fourth chapter,we clear the overall thinking of the thesis framework,and given a brief explanation of the direction to be tried in the future.