An Inverse Problem For A Parabolic Equation Based On Total Variation Regularization Method

In the paper,we studied an inverse problem of finding an unknown parameter of a parabolic equation.The most difficult of the inverse problem is to deal with the ill posedness.The main method of solving the ill posed problem is the Tikhonov regularization method.It was first proposed by a Soviet scientist A.N.Tikhonov in 1960 s.thus put the method of solving the ill posed problem into a new era.But the Tikhonov regularization method requires the solution has a certain smoothness and use it to approximate the solution of the original problem.But in the physical model the solutions are often not smooth enough(such as preserving the image boundary in the process of image restoration),according to the conditions how to solving this problem is the article mainly focus of.This article based on the optimal control theory and using the total variation regularization method,first we introduced a necessary conditions of the inverse problem,then proved uniqueness and stability,at last the numerical solution of the unknown coefficient is reconstructed successfully.This paper mainly includes the following four parts:The first chapter is the introduction.Firstly,it briefly introduced the development history,the research background and the present situation of the research.Secondly,we introduced the main work of this paper.In the second chapter,we dealt with an inverse problem of identifying an unknown coefficient from the angle of theoretical analysis.we used the total variation regularization method for the problem P,then from the optimal control framework the original problem is transformed into an optimal control problem Q,later we deduced the necessary condition of the minimal element.Finally,on the basis of necessary conditions,we proved the stability and uniqueness of the minimizer.The third chapter is numerical simulation.Based on the theoretical analysis of the second chapter,we designed a total variation iterative algorithm and simulated in MATLAB.This chapter can divided into two sections,the first section is the design of the algorithm,and the second section is the numerical experiment.In the second section we also compared with the traditional Landweber iterative algorithm.The fourth chapter is summary and outlook.The main research results of this paper are summarized,and the prospect of further research in this field is made.Since the problem P considered in this paper is a one-dimensional parabolic system,and for the case of highdimension it is usually applied to a wider range of practical applications.So,the study of high dimensional system is an important part of our work in the future.Secondly,the algorithm designed in this paper has a higher sensitivity to the regularization parameter,so how to design a more effective and simple numerical algorithm is also an aspect of our following work.