The Multiscale Method With Matrix Compression Strategy For Solving Iteration Equation With Alternating Direction

The thesis is divided into two parts.In the first part,we study the multiscale Galerkin projection algorithm for solving the iterative equation with alternating direction in Hilbert space.We use the Galerkin projection method with matrix compression to discrete the iterative equation with alternating direction,we prove the approximate solution based on the prior parameter selection strategy and the adaptive parameter selection strategy has optimal convergence order.In the second part,we discuss the multiscale collocation method to solve the Fredholm integral equation of first kind with sectorial compact operator in Banach space.The optimal convergence order of the approximate solution based on the iterative stopping criterion is proved.This part continues the research work of predecessor.The first chapter is the introduction,the numerical methods for solving the first kind of Fredholm integral equation at home and abroad are briefly introduced.In particular,we introduces the source of ill-posed integral equation with compact operator and current research achievements,then briefly reviews the history of development of multiscale fast algorithm.Finally,the main work of this thesis is introduced.In the second chapter,the multiscale projection algorithm for solving the linear Fredholm integral e-quation of the first kind in Hilbert space is studied.We apply the compact integral operator with vanishing moments in the wavelet basis,the multiscale Galerkin projection algorithm with matrix compression is pro-posed to solve the iterative equation with alternating direction.It is proved that the approximate solution based on the prior parameter selection strategy and the adaptive parameter selection strategy has the optimal convergence order.In the third chapter,we study the fast multiscale collocation algorithm for the first kind of the Fredholm integral equation with sectorial operator in Banach space.The previous work has been popularized.We apply the theoretical framework of multiscale fast collocation algorithms in Banach space,present the fast multiscale collocation algorithm with matrix compression to solve the iterative equations with alternating direction.The calculation of non-zero elements of coefficient matrix is reduced.The posterior iteration stopping criterion is given to ensure the optimal convergence order of the approximate solution.In the fourth chapter,we summarizes the advantages and disadvantages of this thesis,it also gives a brief description of the problems and directions to be studied in the future.