Arnoldi Iterative Regularization Method For Discrete Ill-Posed Problems And Its Application

This thesis considers the effective approximate solutions of the following large-scale least squares problems:where, A ? Rn×n, b ? Rn, the singular values of A quickly decay and cluster at zero with no evident gap between two consecutive ones to indicate numerical rank; in particular, A is ill-conditioned and sigular. Minimization problems above with a matrix of this kind are commonly referred to as discrete ill-posed problems. They rise, for instance, derive from the discretization of ill-posed problem such as the first kind of Fredholm integral equation with smooth kernel, and vector b in this problem is generally expressed as known vector data obtained by measurement in scientific and engineering applications.It is usually meaningless to solve this problem because of ill-condition of A and the noisy e in vector b. Therefore, one often replaces the descrete ill-posed problem by a nearby problem, whose solution is less sensitive to the error in b than the solution of the descrete ill-posed problem. This replacement is known as regularization. The possibly most popular regularization method is due to Tikhonov. The paper presents some Arnoldi iterative regularization methods and its application. The contents include three parts:some exsiting Arnoldi iterative methods are systemly summarized; then a new range-restricted Arnoldi iterative method, as well as a generalized Arnoldi iterative method is presented. At the same time, the applications of both metods in Fredholm integral equation of the first kind and image restoration are considered.This thesis consists of five parts. The background and the significance of topic selection, the research progress of this topic, the content and the innovation of the paper are introduced in chapter one. A Lanczos bidiagonalization method and Arnoldi iterative regularization method are introduced in chapter two. A range-restricted Arnoldi iterative method basied on the krylov space, as well as the application in Fredholm integral of the first kind and image restoration, is presented in chapter three. A generalized Arnoldi regularization is obtained by the extended Arnoldi iterative regularization. The application of this method in Fredholm integral of the first kind and image restoration is studied in chapter four. Finally some conclusions are drawn in chapter five.