To Solve The First Kind Of Positive Semidefinite Truncation Fast Algorithm Of Pathological Integral Equation

Most of the inverse problem in mathematical physics can be attributed to the ill-posed integral equation of the first kind, in order to obtain stable approximate solutions, some regularization techniques are required. But regularization equation is an infinite-dimensional system, from view of the numerical solution, always keep the problem of infinite dimension discrete for finite dimension.The main problem encountered in the discrete process is a huge amount of computation.Therefore, under the premise of maintaining the optimal convergence rate, fast solution of the equation is particularly important, and this is a research hotspot in recent years. In fact, the regularization method is effective also depends on the regularization parameter selection. Direct to the semi-definite ill-posed integral equation of the first kind, using the optimization of projection method in this paper,full text is divided into four chapters.The first chapter briefly describes the concept of ill-posed problems, the definition of the first kind Fredholm integral equation, and the main results of this paper are also outlined.Chapter2introduced several important regularization methods and regularization parameter selection strategy.In chapter3, a fast truncated Lavrentiev method is established for solving the semi-definite ill-posed integral equation based on the optimization of projection method. We proposed a priori error estimates and a new posteriori parameter choice strategy. Compared with the traditional projection technique, we obtain the same optimal convagence rate, but less than the number of inner products caculation.In chapter4, the dynamical system method for solving semi-definite integral equation of the first kind is constructed. A priori error estimates is given and a new posteriori parameter choice strategy is proposed, which allows us to reach an optimal order of accuracy.