Regularization Theory And Algorithm For Some Inverse Problems For Parabolic Partial Differential Equations

Some inverse problems for parabolic partial differential equations are studied in this thesis. These inverse problems include inverse heat conduction problems,backward heat conduction problems in time , identification problems for unknown source. We analyze the the essence and degree of these ill-posed problems and provide many regularization methods, especially we focus on the spectral regularization methods,wavelet dual least squares method and finite difference method, prove the error estimates for these methods as well. According to the general theory of regularization, many error estimates are order optimal.This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.