The Application Of Trust Region Algorithm In Solving The Equations Of The First Kind

A trust region algorithm is a new developed algorithm for solving ill-posed problems, and solving the first kind Fredholm integral equation is a special kind of inverse problem. Ill-posedness is the characteristics of this problem. In order to obtain a stable numerical solution, regularization method must be used.The main idears of this study are using trust region algorithms for solving first kind Fredholm integral equations. For solving with weakly singular Fredholm integral equation kernel and general two-dimensional, three-dimensional first kind Fredholm integral equation has always been an important topic in research fields, and used trust region algorithm for solving such problems is the focus of this research.First, we gave the basic model of Fredholm integral equations and applications and described the difficulties of the problem-solving; Secondly, the system description of trust region algorithm, and gives convergence analysis;With trust region algorithms solving and discreting for weakly singular kernel of Fredholm integral equation and general two-dimensional, three-dimensional first kind Fredholm integral equation and with the calculation results of Tikhonov regularization method and TSVD Regularization Method compared.The results of numerical simulation and experimental analysis verify that paper gives the trust region algorithm with a one-dimensional weakly singular kernel Fredholm integral equation of the first category is feasible; solving the kernel of non-singular two-dimensional and three-dimensional first kind Fredholm integral equation depends on the kernel changes and smoothness of true solution. In addition, there are also a relationship with disturbance of data grid and subdivision.