Dynamical Systems Method For Solving Ill-Posed Problems

In rencent decades, the subject of Mathematical physics inverse problem has achieved a rather rapid development. The development of such subject, to a great degree, is driven by the urgent needs which come from the applications in other subjects and numerous Engineering technologies. Mathematical physics inverse problem is no longer the pure inverse problem in Mathematics and Physics. Due to the development of science and technology and enlargement of research scale, Geology, Image, Remote sensing, Petroleum exploration, Medical science, Finance, Economy and even life science have proposed inverse problem of exploring reason(undetermined inverse parameter) from result (observation). Thus, inverse problem possess the characteristics of a wide covering range, rich contents, cross-industry, cross-subject。Seeing from the research method of inverse problem, it covers more knowledge in Computational Mathematics, applied mathematics and statistics. We can say that the theory and method in Mathematics is the basis of the study of inverse problem. In the history of the inverse problem, it represented one of the most active and encouraging interdisciplines. Most inverse problems are Ill-posed.There are lots of methods to solve ill-posed problems. This paper is mainly use of Dynamical System Method to solve the first kind of ill-posed problems. According to the property of the operators, we divide the equations into linear and nonlinear, and discuss them respectively. For the linear operator equations, it can be written as Au=f, where A is a linear operator in Hilbert spaces H. We use the Dynamical System Method and Regularization method to solve the solution of the regularization of this problem. According the semi-group theory of the linear operator, the regularization equation u'(t)＝-A*(Au(t)-f) of the problem has been solved. With linear operator semigroup theory, we get the solution of the regularization equation, and prove that the regularized solution converged to the solution of the original problem when t�'∞.For the nonlinear operator equations, it can be written as F(x)=y, where F:D(F)(?)X�'Y is a nonlinear differentiable operator between the Hilbert spaces X and Y, whose Frechet derivative F'(u) is locally uniformly bounded. Using the Dynamical System Method and Regularization method, the regularization equation u'(t)=F'(u(t))*(yδ-F(u(t))),t≥0,u(0)=x0 of the problem has been solved.Because the regularized solution of nonlinear equation cannot be obtained directly, we need discrete it and transform it into exponential Euler: u(n+1)=un+hnφ(-hnJ(un))F'(un)*(yδ-F(un))。where J(u)=F'(u)*F'(u),φ=e(?), and prove that the regularized solution converged to the solution of the original problem when n�'∞.