The Study Of Dynamical System Methods For Nonlinear Ill-Posed Problems

Dynamical system methods, which can also be called continuous regularizationmethod, are very effective methods for solving nonlinear ill-posed problems. Thesemethods overcome the stronger restrictive conditions on the operator in the conver-gence theorems known for the corresponding iterative methods, and prove the stabilityof the systems which we investigated. It is well known that the theory of dynamicalsystems and their stability have been remarkable issue. The Lyapunov theory, whichare important techniques for analyzing the stability of the systems, obtains furtherdevelopment, and becomes to be perfect in this paper. We will investigate the nonlin-ear ill-posed problems by means of the idea of dynamical systems and the Lyapunovtheory in some aspects.Dynamical systems can be divided into two kinds of systems, that is, one is adiscrete system, and the other is continuous. Firstly, we set about to study the dis-crete one. Based on the continuous Landweber method, we construct a Runge–Kutta(simply as R–K) type Landweber method for solving nonlinear ill-posed problems,and investigate the convergent property of this method. Furthermore, we obtain theconvergence rate of this method when the perturbed data with noise exists. Comparedwith the numerical performance of Landweber method, the convergence rate of R–Ktype method is higher, and the method is more stable.When the operators are unbounded we get ride of the Fr′echet differentiability ofthe parameter–to–output map as well as conditions restricting its nonlinearity, intro-duce a derivative-free and special structure, and put forward a parameter identification.Convergence and stability to the parameter identification have been proved in theoryunder the more weak restrictive conditions associated with the solvability of the directproblem.Aiming at a minimization problem for solving nonlinear ill-posed problems, wegive a new lemma of Lyapunov stability based on the original Lyapunov stability.Moreover, we use this lemma to prove the convergence of the minimization problem.The restrictive conditions on this new lemma of stability are weaker than those Xuproposed. Therefore, the lemma of stability is a continuation of the original one, and a remarkable innovation in this paper.Since the classical iterative methods for nonlinear ill-posed problems are all lo-cally convergent, we construct a robust and widely convergent homotopy regulariza-tion to identify the parameter in view of the properties of homotopy method, andproved this method to be convergent in the light of the theory of Lyapunov stability.Compared with Landweber iteration, a concrete numerical example proved this ho-motopy regularization to be more stable and widely convergent with the same noise.Based on the solvability and stability of the partial differential equation inSobolev space, we use the similar idea to the dynamical systems and put forwarda level set method for the identification problem of the nonlinear parabolic distributedparameter systems. Moreover, we validate the level set approach to be a regularizationif the discrepancy principle is used as a stopping rule.