Iterative Methods And Application For Inverse Problems

The dissertation studies the iterative methods for general nonlinear inverse problem and a few kinds of linear inverse heat conduction problems. We obtain the convergence and convergence rates of regular solution. At the same time, we introduce a kind of two dimensional inverse heat conduction problems which do not convect with outer space in doing the compression, and we have got the expected result.In the first chapter, we present the basic notion and some applications of inverse problems. We also summarize some important regularization methods.In chapter 2, we consider nonlinear inverse problems in the framework of abstract operator equations of the form F(x)=y. We give a new Newton-Landweber iteration to solve this kind of nonlinear equation. We get the convergence and convergence rate of this iteration under the following source condition x+-x0=(F'(x+)*F'x+))μω. Meanwhile, by introducing the Hilbert scales, we establish a preconditioned Newton-Landweber iteration and prove that this iteration will greatly decrease the iterative steps.In chapter 3, the numerical solution to the problem related to concrete linear inverse problems is considered. This kind of problem is called sideways parabolic equation. Iterative method in the following type is established. We prove that our method is of order optimal under both a priori and a posteriori stopping rules. Furthermore, if we use the discrepancy principle, we can avoid the selection of the a priori bound. An appropriate selection of the parameter n in the iteration scheme helps greatly to reduce the iterative steps and get a satisfactory approximate solution.In chapter 4, we discuss the iterative method of a kind of two dimensional inverse heat conduction problems which do not convect with outer space. We show that these problems are severely ill-posed. Similar to chapter 3, an iterative method has been established and convergence rate has been proved. A quite useful solution model has been deduced. Finally, we apply the inverse heat conduction problem to image compression. Excited results have been got.Abundant numerical tests have been done among the last three chapters. The numerical results support well the theory results. KEY WORDS ill-pose, regularization, convergence, convergence rate, heat conduction problem, image compression...