چكيده به لاتين
A common problem in diverse areas of mathematics and physical sciences is trying to find a point in the intersection of convex sets. This problem is referred to as the convex feasibility problem (CFP).
CFP has a long and rich history in mathematics from 19th century. This problem has diverse applications in mathematics and modern physical sciences such as: Medical imaging and radiation therapy treatment planning(computerized tomography), Electron microscopy, Signal processing, Minimization of convex non smooth functions and etc.
When the convex sets are the linear equations or inequalities, the obtained problem is called a linear feasibility problem (LFP). Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. The image reconstruction problem is a well known example, which can be modeled as a linear feasibility problem by using the Radon transform.
Since the obtained linear systems of equations are typically large, sparse and ill-posed, it calls for the use of iterative methods (against direct methods) for solving these systems.
In this thesis, we study a general block iterative scheme based on Landweber algorithm so that some widely iterative methods such as the ART, CAV, BICAV and CIM are special examples of it.
When applying the block iterative methods based on Landweber algorithm for solving an ill-posed set of linear equations the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. In this thesis, we study the semi-convergence behavior of these methods and obtain an upper bound for data error (noise error). Based on this bound, we propose new ways to specify the relaxation parameters to control the semi-convergence. The performance of our strategies is shown by examples taken from tomographic imaging.
In the first chapter, after the explanation of problem, we give some of the basic definitions and theorems that will be required. In chapter 2, we do short overview of the some of iterative methods used for solving the linear feasibility problems. In chapter 3, we do a careful analysis of the semiconvergence behavior of the block iterative methods and propose the new techniques to control this phenomenon. In chapter 4, we present an extended form of the block iterative methods which covers the existing block iterative methds. Furthermore it allows us posibility of using more iterative methods. In chapter 5, we consider a family of the block iterative methods based on column-partitioning
of coefficient matrix for solving linear system of equations and analyze its convergence. In this family, relaxation parameters and weight matrices can be updated in each iteration. Furthermore, we are allowed to change the blocks-column-partitioning in each cycle.
Keywords: The Landweber algorithm, Block iterative methods, Semi convergence, Image reconstruction
